1 Introduction

Let \(K \subset \mathbb R ^d\) be a compact convex body with non-empty interior and having a \(C^3\) boundary of positive Gaussian curvature \({\kappa }\). Letting \(\mathcal{P }_{\lambda }\) be a Poisson point process in \(\mathbb R ^d\) of intensity \({\lambda }, {\lambda }\in (0, \infty ),\) we denote by \(K_{\lambda }\) the convex hull of \(\mathcal{P }_{\lambda }\cap K\). Let \(f_k(K_{\lambda }), k \in \{0,1,\ldots ,d-1\},\) be the number of \(k\) faces of \(K_{\lambda }\).

Rényi and Sulanke [16] were the first to consider the average behavior of \(f_0(K_{\lambda })\) in the planar case. Generalizing their formula to higher dimensions, Bárány [1] showed there is a constant \(D_{0,d}\) such that

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-(d-1)/(d + 1)} \mathbb E f_0(K_{\lambda }) = D_{0,d} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$

The integral \(\int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz\) is known as the affine surface area of the boundary \({ \partial K }\). Assuming only that \(\partial K\) is of differentiability class \(C^2\), Reitzner [15] extended this result to \(f_k(K_{\lambda }), \ k \in \{0,1,\ldots ,d-1\}\), showing for all \(d \ge 2\) that there are constants \(D_{k,d}\) such that

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-(d-1)/(d + 1)} \mathbb E f_k(K_{\lambda }) = D_{k,d} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$
(1.1)

Reitzner [14] also showed that \((f_k(K_{\lambda })- \mathbb E f_k(K_{\lambda }))/\sqrt{ \mathrm{Var}f_k(K_{\lambda }) }\) converges in distribution to a mean zero normal random variable as \({\lambda }\rightarrow \infty \), though there have been relatively few results concerning the asymptotic variance of \(f_k(K_{\lambda })\). Theorem 4 of Reitzner [14] gives upper and lower bounds of the same magnitude for \(\mathrm{Var}f_k(K_{\lambda }), k \in \{0,1,\ldots ,d - 1\},\) which extends work of Buchta [7], who obtains lower bounds for \(\mathrm{Var}f_0(K_{\lambda })\) of order \({\lambda }^{(d-1)/(d + 1)}\). In the special case that \(K\) is a ball, closed form variance asymptotics for \(\mathrm{Var}f_k(K_{\lambda }), k \in \{0,1,\ldots ,d - 1\}\) are given in [8, 19].

Let \(K_n^{\prime }\) be the convex hull of \(n\) i.i.d. random variables uniformly distributed on \(K\). Our main two results resolve the open question of determining variance asymptotics for \(\mathrm{Var}f_k(K_{\lambda })\) and \(\mathrm{Var}f_k(K_n^{\prime })\), \(K\) smooth and convex, as put forth on p. 1431 of [21]. For all \(m = 1, 2,\ldots \), let \(\mathcal{K }^m_+\) be the collection of compact convex sets \(K\) in \(\mathbb R ^d\) with boundary of class \(C^m\) and having positive Gaussian curvature.

Theorem 1.1

For all \(k \in \{0,1,\ldots ,d-1\}\), there exists a constant \(F_{k,d} \in (0, \infty )\) such that for all \(K \in \mathcal{K }^3_+\)

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-(d-1)/(d + 1)} \mathrm{Var}f_k(K_{\lambda }) = F_{k,d} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$
(1.2)

Let \(\mathrm {vol}\) denote Lebesgue measure on \(\mathbb R ^d\). De-Poissonization methods, based on coupling, yield the following binomial counterpart of (1.2). When \(k = 0\), it resolves Conjecture 1 of Buchta [7].

Theorem 1.2

For all \(k \in \{0,1,\ldots ,d-1\}\) and \(K \in \mathcal{K }^3_+\)

$$\begin{aligned} \lim _{n \rightarrow \infty } n^{-(d-1)/(d + 1)} \mathrm{Var}f_k(K_n^{\prime }) = F_{k,d}(\mathrm{{vol}}(K))^{-(d-1)/(d+1)} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz.\nonumber \\ \end{aligned}$$
(1.3)

Remark

  1. (i)

    Related work. Bárány and Reitzner [4, p. 3] conjecture for general convex bodies that \(\mathrm{Var}f_k(K_{\lambda })\) should, up to constants, behave like the variance of the volume of the wet part of the floating body, which, in the case of smooth convex sets, is proportional to affine surface area. Theorem 1.1 resolves a sharpened version of this conjecture in the case that \(\partial K\) is smooth.

  1. (ii)

    The constants \(F_{k,d}\). The proofs of Theorems 1.1 and 1.2 show that \(F_{k,d}\) is defined in terms of parabolic growth processes on \(\mathbb R ^{d-1} \times \mathbb R ^+\), a fact first recognized by Tomasz Schreiber in the context of \(K = \mathbb{B }^d\). As noted on p. 137 of Buchta [7], \(F_{k,2}\) may also be identified in terms of a constant involving complicated double integrals given in Groeneboom [9].

Theorems 1.1 and 1.2 yield asymptotics for \(\mathrm{Var}\; \mathrm{vol }(K_n^{\prime })\) and \(\mathrm{Var}\; \mathrm{vol }(K_{\lambda })\), which goes as follows. Recall that under \(C^3\) and \(C^2\) assumptions on \({ \partial K }\), respectively, Bárány [1] and Reitzner [13] show

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{2/(d + 1)} \mathbb E \mathrm{{vol}}(K \setminus K_{\lambda }) = c_{d} (\mathrm{{vol}}(K))^{2/(d + 1)} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$
(1.4)

Böröczky et al. [6] extend this limit and (1.1) to convex hulls of i.i.d. points having a non-uniform density on \(K\). Theorem 3 of Reitzner’s breakthrough paper [14] gives upper and lower bounds of the same magnitude for \(\mathrm{Var}\; \mathrm{{vol}}(K_{\lambda })\), though it falls short of prescribing a limiting variance. Theorems 1.1 and 1.2 fill in this gap as follows. When \(K \in \mathcal{K }^{d + 6}_+\), Buchta notes (see Corollary 1 and (3.6) of [7]) that variance asymptotics for \(n^2 \mathrm{Var}f_0(K_n^{\prime })\) and \(\mathrm{Var}\; \mathrm{ vol }(K_n^{\prime })\) coincide, since

$$\begin{aligned} \mathrm{Var}\; \mathrm{{vol}}(K_n^{\prime })=\frac{\mathrm{Var}(f_0(K_{n + 2}^{\prime }))+d_{n+2}}{(n+1)(n+2)}, \end{aligned}$$

where \(d_n, n \ge 1,\) satisfies

$$\begin{aligned} \lim _{n \rightarrow \infty }\left(\frac{3-d}{d+1}\int _{\partial K}\kappa (z)^{1/(d + 1)}dz\cdot n^{(d-1)/(d+1)}\right)^{\!-1}d_n=1. \end{aligned}$$

Consequently, putting \(k = 0\) in (1.3) and \(G_d:=F_{0,d}+(3-d)/(d+1)\) we get the following corollary.

Corollary 1.1

For all \(K \in \mathcal{K }^{d + 6}_+\) we have

$$\begin{aligned} \lim _{n \rightarrow \infty } n^{(d+3)/(d + 1)} \mathrm{Var}\; \mathrm{vol }(K_n^{\prime }) = G_{d}(\mathrm{{vol}}(K))^{(d+3)/(d+1)} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz.\nonumber \\ \end{aligned}$$
(1.5)

By (1.5) and Proposition 3.2 of [20], which states that \(\mathrm{Var}\;\mathrm{{vol}}(K_n^{\prime })\) and \(\mathrm{Var}\;\mathrm{{vol}}(K_n)\) coincide up to first order, we deduce the next result.

Corollary 1.2

For all \(K \in \mathcal{K }^{d + 6}_+\) we have

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \lambda ^{(d+3)/(d + 1)} \mathrm{Var}\; \mathrm{vol }(K_{\lambda }) = G_{d} \int _{\partial K} {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$
(1.6)

The paper is organized as follows. Section 2 introduces the main tool for the proof of Theorem 1.1, namely the paraboloid growth process used in [19] and [8]. We state a general result, Theorem 2.1, giving expectation and variance asymptotics for the empirical \(k\)-face measure, which includes Theorem 1.1 as a special case. Theorem 2.1 also shows that the constants \(F_{k,d}\) of Theorem 1.1 involve integrals of one and two point correlation functions of a scaling limit \(k\)-face functional \(\xi _k^{(\infty )}\) associated with parabolic growth processes. Section 3 introduces an affine transform of \(K\) and a scaling transform of the affine transform to link the finite volume \(k\)-face functional with the infinite volume scaling limit \(\xi _k^{(\infty )}\). Section 4 contains the main technical aspects of the paper, focussing on properties of re-scaled \(k\)-face functionals. In particular Lemmas 4.5 and 4.8 show that the one and two point correlation functions of the re-scaled \(k\)-face functional on the affine transform of \(K\) are well approximated by one and two point correlation functions of the re-scaled \(k\)-face functional on an osculating ball. In this way the expectation and variance asymptotics for \(f_k(K_{\lambda })\), \(K\) an arbitrary smooth body, are controlled by the corresponding asymptotics for \(f_k(K_{\lambda })\) when \(K\) is a ball. The latter asymptotics are established in [8]. Section 5 contains the proof of Theorem 2.1 and Section 6 establishes the de-Poissonized limit (1.3).

2 Paraboloid growth processes and a general result

Given a finite point set \(\mathcal{X }\subset \mathbb R ^d\), let co\((\mathcal{X })\) be its convex hull.

Definition 2.1

Given \(k \in \{0,1,\ldots ,d-1\}\) and \(x\) a vertex of co\((\mathcal{X })\), define the \(k\)-face functional \(\xi _k(x, \mathcal{X })\) to be the product of \((k +1)^{-1}\) and the number of \(k\) faces of \(\text{ co}(\mathcal{X })\) which contain \(x\). Otherwise we put \(\xi _k(x, \mathcal{X }) = 0\). The empirical k-face measure is

$$\begin{aligned} \mu ^{\xi _k}_{\lambda }:= \sum _{x \in \mathcal{P }_{\lambda }\cap K} \xi _k(x, \mathcal{P }_{\lambda }\cap K) {\delta }_x, \end{aligned}$$
(2.1)

where \({\delta }_x\) is the unit point mass at \(x\).

Thus the number of \(k\)-faces in \(\text{ co}(\mathcal{X })\) is \(\sum _{x \in \mathcal{X }} \xi _k(x, \mathcal{X })\). We shall give a general result describing the limit behavior of \(\mu ^{\xi _k}_{\lambda }\) in terms of parabolic growth processes on \(\mathbb{R }^d\).

Paraboloid growth processes. Denote points in \(\mathbb R ^{d-1} \times \mathbb R \) by \( w:= (v,h)\) or \( w^{\prime }:= (v^{\prime },h^{\prime })\), depending on context. Let \(\Pi ^{\uparrow }\) be the epigraph of the parabola \(v \mapsto |v|^2/2\), that is \(\Pi ^{\uparrow } := \{(v,h) \in \mathbb R ^{d-1} \times \mathbb R ^+, h \ge |v|^2/2\}.\) Letting \(\mathcal{X }\subset \mathbb R ^d\) be locally finite, define the parabolic growth model

$$\begin{aligned} \Psi (\mathcal{X }):= \bigcup _{ w \in \mathcal{X }} (w \oplus \Pi ^{\uparrow }), \end{aligned}$$

where \(\oplus \) denotes Minkowski addition. A point \(w_0 \in \mathcal{X }\) is extreme with respect to \(\Psi (\mathcal{X })\) if the epigraph \(w_0 \oplus \Pi ^{\uparrow }\) is not a subset of the union of the epigraphs \(\{w \oplus \Pi ^{\uparrow }, w \in \mathcal{X }\setminus w_0\}\), that is \((w_0 \oplus \Pi ^{\uparrow }) \nsubseteq \bigcup _{w \in \mathcal{X }\setminus w_0} (w \oplus \Pi ^{\uparrow }).\)

The paraboloid hull model \( \Phi (\mathcal{X })\) is defined as in Definition 3.4 of [8]:

$$\begin{aligned} \Phi (\mathcal{X }):=\bigcup _{\left\{ \begin{array}{c} w\in \mathbb R ^{d-1}\times \mathbb R \\ (w \oplus {{\Pi }^{\downarrow }})\cap \mathcal{X }=\emptyset \end{array}\right.} (w\oplus {{\Pi }^{\downarrow }}), \end{aligned}$$

where \(\Pi ^{\downarrow }:= \{(v,h) \in \mathbb R ^{d-1} \times \mathbb R , h \le - |v|^2/2\}.\) It may be viewed as the dual of the paraboloid growth model \(\Psi (\mathcal{X })\). Let \(\mathcal{P }\) be a rate one homogeneous Poisson point process on \(\mathbb R ^{d-1} \times \mathbb R ^+\) and let \(\Psi := \Psi (\mathcal{P })\) and \(\Phi := \Phi (\mathcal{P })\) be the corresponding paraboloid growth and hull processes. As in [8], the set \(\mathrm{Vertices}(\Phi )\) coincides with the extreme points of \(\Psi \). We recall (cf. Definition 3.3 of [8]) that a set of \((k+1)\) extreme points \(x_1,\ldots ,x_{k+1}\), \(k\ge 0\), generates a so-called \(k\)-dimensional paraboloid face if there exists a translate \(\tilde{\Pi }^{\downarrow }\) of \(\Pi ^{\downarrow }\) such that \(\{x_1,\ldots ,x_{k+1} \}= \tilde{\Pi }^{\downarrow } \cap \mathcal{X }\).

Definition 2.2

(cf. section 6 of [8]) Define the scaling limit \(k\)-face functional \(\xi ^{(\infty )}_{k}(x, \mathcal{P })\), for \(x \in \mathcal{P }\), and \(k \in \{0, 1,\ldots , d - 1\}\), to be the product of \((k + 1)^{-1}\) and the number of \(k\)-dimensional paraboloid faces of the hull process \(\Phi \) which contain \(x\), if \(x\) belongs to \(\mathrm{Vertices}(\Phi )\), and zero otherwise.

One of the main features of our approach is that \(\xi ^{(\infty )}_{k}, k \in \{0,1,\ldots ,d-1\},\) are indeed scaling limits of appropriately re-scaled \(k\)-face functionals, as seen in Lemmas 4.6 and 4.7 of Sect. 4.

Define the following second order correlation functions for \(\xi ^{(\infty )}(x, \mathcal{P }) \!\!\!:=\xi ^{(\infty )}_{k}(x, \mathcal{P })\) (cf. (7.2), (7.3) of [8]).

Definition 2.3

For all \(w_1, w_2 \in \mathbb R ^d\), put

$$\begin{aligned} \zeta _{\xi ^{(\infty )}}(w_1,w_2)&:= \zeta _{\xi ^{(\infty )}}(w_1, w_2, \mathcal{P })\nonumber \\&:= \mathbb E \xi ^{(\infty )}(w_1, \mathcal{P }\cup \{w_2\} ) \xi ^{(\infty )}(w_2, \mathcal{P }\cup \{w_1\} ) \nonumber \\&- \mathbb E \xi ^{(\infty )}(w_1, \mathcal{P }) \mathbb E \xi ^{(\infty )}(w_2, \mathcal{P }). \end{aligned}$$
(2.2)

Note that

$$\begin{aligned} \sigma ^2(\xi ^{(\infty )}) := \int _0^{\infty } \mathbb E \xi ^{(\infty )}((\mathbf{0},h), \mathcal{P })^2 dh + \int _0^{\infty } \int _\mathbb{R ^{d-1}} \int _0^{\infty } \varsigma _{\xi ^{(\infty )}}((\mathbf{0},h),(v^{\prime },h^{\prime })) dh^{\prime } dv^{\prime } dh\nonumber \\ \end{aligned}$$
(2.3)

is finite and positive by Theorems 7.1 and 7.3 in [8].

Theorem 1.1 is a special case of the following general result expressing the asymptotic behavior of the empirical \(k\)-face measures in terms of parabolic growth processes. Let \(\mathcal C (K)\) be the class of continuous functions on \(K\) and let \(\langle g, \mu _{\lambda }^\xi \rangle \) denote the integral of \(g\) with respect to \(\mu _{\lambda }^\xi \).

Theorem 2.1

For all \(k \in \{0,1,\ldots ,d-1\}\), \(K \in \mathcal{K }^3_+\), and \(g \in \mathcal C (K)\) we have

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-(d-1)/(d + 1)} \mathbb E [\langle g, \mu _{\lambda }^{\xi _k} \rangle ] = \int _0^\infty \mathbb E \xi ^{(\infty )}_{k}((\mathbf{0},h), \mathcal{P }) dh \int _{{ \partial K }} g(z) {\kappa }(z)^{1/(d + 1)} dz\nonumber \\ \end{aligned}$$
(2.4)

and

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-(d-1)/(d + 1)} \mathrm{Var}[\langle g, \mu _{\lambda }^{\xi _k} \rangle ] = \sigma ^2(\xi ^{(\infty )}_{k}) \int _{{ \partial K }} g(z)^2 {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$
(2.5)

Remarks (i) Related work. Up to now, (2.5) has been known only for bodies of constant curvature, i.e., only for \(K = r\mathbb{B }^d, d \ge 2, r > 0\); see Theorem 7.3 of [8].

(ii) The constants. The convergence (1.2) is implied by (2.5) with \(F_{k,d}=\sigma ^2(\xi ^{(\infty )}_{k})\). Indeed, applying (2.1) to \(g \equiv 1\), we have

$$\begin{aligned} \langle 1, \mu _{\lambda }^{\xi _k} \rangle =\sum _{x\in \mathcal{P }_{{\lambda }}\cap K}\xi _k(x, \mathcal{P }_{\lambda }\cap K) =f_k(K_{{\lambda }}). \end{aligned}$$

(iii) Extensions. As in [14] and [6], we expect that our main results hold for \(K \in \mathcal{K }^2_+\) and we comment on this at the end of Sect. 5.2. Following the methods of Sect. 6, we may obtain the counterpart of Theorem 2.1 for binomial input.

3 Affine and scaling transformations

Fix \(K \in \mathcal{K }^3_+\). For each \(z \in \partial K\), we first consider an affine transformation \(\mathcal{A }_z\) of \(K\), one under which the scores \(\xi _k\) are invariant, but under which the principal curvatures of \(\mathcal{A }_z(K)\) at \(z\) coincide, that is to say \(\mathcal{A }_z(K)\) is ‘umbilic’ at \(z\). This property allows us to readily approximate the functionals \(\xi _k\) on Poisson points in \(\mathcal{A }_z(K)\) by the corresponding functionals on Poisson points in the ‘osculating ball’ at \(z\), defined below. The key idea of replacing the mother body \(K\) with an osculating ball has been used by Rényi and Sulanke [17], Bárány [1], and Böröczky et al. [6], among others.

We in turn transform \(\mathcal{A }_z(K)\) to a subset of \(\mathbb R ^{d-1} \times \mathbb R \) via scaling transforms \(\mathcal T ^{{\lambda }, z},{\lambda }\ge 1\). These transforms yield re-scaled \(k\)-face functionals \(\xi ^{{\lambda }, z}\) on the Poisson points \(\mathcal T ^{{\lambda }, z}( \mathcal{P }_{\lambda }\cap \mathcal{A }_z(K))\), ones which are well approximated by re-scaled \(k\)-face functionals on the image under \(\mathcal T ^{{\lambda }, z}\) of Poisson points in the osculating ball at \(z\). In the large \({\lambda }\) limit the latter in turn converge to the scaling limit functionals \(\xi ^{(\infty )}\) given in Definition 2.2.

In this way the expectation and variance asymptotics for \(k\)-face functionals on Poisson points in \(K\) are obtained by averaging, with respect to all \(z \in \partial K\), the respective asymptotics for the re-scaled \(k\)-face functionals on Poisson points in osculating balls at \(z\). The limit theory of the latter is established in [8, 19] and we shall draw upon it in our approach.

3.1 Affine transformations \(\mathcal{A }_z, z \in K\)

Fix \(K \in \mathcal{K }^3_+\). Let \(\mathcal{M }(K)\) be the medial axis of \(K\), i.e. the set of points \(x\) in the interior of \(K\) such that there exists a maximal ball centered at \(x\) and included in \(K\). In other words, \(\mathcal{M }(K)\) is the set of interior points having more than one closest point on \(\partial K\). \(\mathcal{M }(K)\) has Lebesgue measure zero and we parameterize points \(x \in K \setminus \mathcal{M }(K)\) by \(x := (z, t)\), where \(z \in \partial K\) is the unique boundary point closest to \(x\) and where \(t \in [0, \infty )\) is the distance between \(x\) and \(z\).

Denote by \(C_{z,1},\ldots ,C_{z,d-1}\) the principal curvatures of \(\partial K\) at \(z\), i.e. the eigenvalues of the Weingarten operator at \(z\). Let \({\kappa }(z): = \prod _{i=1}^{d - 1} C_{z,i}\) be the Gaussian curvature at \(z\), so that the Gaussian curvature radius \(r_z\) satisfies \(\kappa (z)= r_z^{-(d-1)}\).

For \(z \in \partial K\), consider the affine transformation \(\mathcal{A }_z\) which preserves \(z\), the Lebesgue measure, the unit inner normal to \(z\), and which transforms the Weingarten operator at \(z\) into \(r_z^{-1}I_{d-1}\) where \(I_{d-1}\) is the identity matrix of \(\mathbb R ^{d-1}\). Under the action of \(\mathcal{A }_z\), the number of \(k\)-faces of the random convex hull inside the mother body \(K\) is preserved. Additionally, \(\xi _k, k \in \{0,1,\ldots ,d-1\}\) is stable under the action of \(\mathcal{A }_z\), namely

$$\begin{aligned} \xi _k(x, \mathcal{P }_{\lambda }\cap K) = \xi _k(\mathcal{A }_z(x), \mathcal{A }_z(\mathcal{P }_{\lambda }\cap K)). \end{aligned}$$
(3.1)

Indeed, \(\mathcal{A }_z\) sends any \(k\)-face of \(K_{{\lambda }}\) to a \(k\)-face of \(\mathcal{A }_z(K_{{\lambda }})\). This follows since affine transformations preserve convexity and convex hulls. A \(k\)-face \(F_k \) of \(K_{{\lambda }}\) is a.s. the convex hull of \((k+1)\) points from \(\mathcal{P }_{\lambda }\), so it is sent to the convex hull of the images by \(\mathcal{A }_z\). Moreover, any support hyperplane \(H\) such that \(H\cap K_{{\lambda }}=F_k\) is sent to a support hyperplane of the image of \(K_{{\lambda }}\) such that its intersection with it is the image of the face \(F_k\). So the image of \(F_k\) is also a \(k\)-face of the image of \(K_{{\lambda }}\).

Put \(K_z:= \mathcal{A }_z (K)\). By construction the principal curvatures at \(z\) all equal \(r_z^{-1}\). We recall that \(\mathcal{A }_z\) preserves the distribution of \(\mathcal{P }_{\lambda }\) so in the sequel, we shall make a small abuse of notation by identifying \(\mathcal{P }_{{\lambda }}\) and \(K_{{\lambda }}\) with \(\mathcal{A }_z(\mathcal{P }_{{\lambda }})\) and \(\mathcal{A }_z(K_{{\lambda }})\), respectively. Let \(B_r(x)\) denote the Euclidean ball of radius \(r\) centered at \(x\). Define the osculating ball at \(z \in \partial K\) to be the ball whose center \(z_0:=z_0(z)\) is at distance \(r_z\) from \(z\) along the inner normal to \(z\). Lemma 4.4 shows that the boundary of the osculating ball \(B_{r_z}(z_0)\) is not far from \(\partial K_z\), justifying the terminology.

Given \(z \in \partial K\), define \(f:\mathbb{S }^{d-1}\mapsto \mathbb R ^+\) to be the function such that for all \(u\in \mathbb{S }^{d-1}\), \((z_0+f(u)u)\) is the point of the half line \((z_0+\mathbb R ^+u)\) contained in \(\partial K_z\) and furthest from \(z_0\). Thus \(\partial K_z\) is given by \(( f(u),u), u \in \mathbb{S }^{d-1}\). Given \(z \in \partial K\) we let the inner unit normal be \(k_z:={ (z_0-z) / |z-z_0| }\). Here and elsewhere we let \(|w|\) denote the Euclidean norm of \(w\). For each fixed \(z \in { \partial K }\), we parameterize points \(w\) in \(\mathbb R ^{d-1} \times \mathbb R \) by \((r, u)\) where \(r := |w - z_0|\) and where \(\in \mathbb{S }^{d-1}\). Henceforth, points \((r,u)\) are with reference to \(z\). For \(z=(r_z,u_z) \in { \partial K }\) (where \(u_z=-k_z\)), let \(T_z \sim \mathbb R ^{d-1}\) denote the tangent space to \(\mathbb{S }^{d-1}\) at \(u_z\). The exponential map \(\exp _{d-1}: T_z \rightarrow \mathbb{S }^{d-1} \) maps a vector \(v\) of the tangent space to the point \(u \in \mathbb{S }^{d-1} \) such that \(u\) lies at the end of the geodesic of length \(|v|\) starting at \(z\) and having direction \(v\). We let the origin of the tangent space be at \(u_z\).

3.2 Re-scaled \(k\)-face functionals \(\mathcal T ^{{\lambda }, z}, z \in \partial K, {\lambda }\ge 1\)

Having transformed \(K\) to \(K_z\), we now re-scale \(K_z\) for all \({\lambda }\ge 1\) with scaling transform denoted \(\mathcal T ^{{\lambda }, z}\). Our choice of \(\mathcal T ^{{\lambda }, z}\) is motivated by the following desiderata. First, consider the epigraph of \(s_{\lambda }: \mathbb S ^{d-1} \mapsto \mathbb R \) defined by

$$\begin{aligned} s_{\lambda }(u, \mathcal{P }_{\lambda }) = r_z - h_{K_{\lambda }}(u), \ u \in \mathbb S ^{d-1}, \end{aligned}$$

where we recall that \(r_z\) is the Gaussian curvature radius at \(z\) and \(h_{K_{\lambda }}(u):= \sup \{ \langle x, u \rangle , \ x \in K_{\lambda }\}\) denotes the support function of \(K_{\lambda }\). Noting that \(h_{K_{\lambda }}(u) = \sup _{x \in \mathcal{P }_{\lambda }} h_x(u)\) for \(u \in \mathbb S ^{d-1}\), it follows that the considered epigraph is the union of epigraphs, which, locally near the vertices of \(K_{\lambda }\), are of parabolic structure. Thus any scaling transform should preserve this structure, as should the scaling limit. Second, a subset of \(K_z\) close to \(z\) and having a unit volume scaling image should host on average \(\Theta (1)\) points of the re-scaled points, that is to say the intensity density of the re-scaled points should be of order \(\Theta (1)\). As in Section 2 of [8], it follows that the transform \(\mathcal T ^{{\lambda }, z}\) should re-scale \(K_z\) in the \((d-1)\) tangential directions with factor \({\lambda }^{1/(d + 1)}\) and in the radial direction with factor \({\lambda }^{2/(d + 1)}\). It is easily checked that the following choice of \(\mathcal{T }^{{\lambda }, z}\) meet these criteria; cf. Lemma 3.1 below. Throughout we put

$$\begin{aligned} {\beta }:= {1\over d + 1}. \end{aligned}$$

Define for all \(z \in { \partial K }\) and \({\lambda }\ge 1\) the finite-size scaling transformation \(\mathcal T ^{{\lambda }, z}: \mathbb R ^+ \times \mathbb{S }^{d -1} \rightarrow \mathbb R ^{d -1} \times \mathbb R \) by

$$\begin{aligned} \mathcal T ^{{{\lambda },z}} \left( {\left( {r,u} \right)} \right): = \left( {\left( {r_{z}^{d} \lambda } \right)^{\beta } \exp _{{d - 1}}^{{ - 1}} \left( u \right),\left( {r_{z}^{d} \lambda } \right)^{{2\beta }} \left( {1 - \frac{r}{{r_{z} }}} \right)} \right): = \left( {v^{^{\prime }}, h^{^{\prime }} } \right): = w^{^{\prime }}.\nonumber \\ \end{aligned}$$
(3.2)

Here \(\exp _{d-1}^{-1}(\cdot )\) is the inverse exponential map, which is well defined on \(\mathbb S ^{d-1} \setminus \{ - u_z \}\) and which takes values in the ball of radius \(\pi \) and centered at the origin of the tangent space \(T_z\). We shall write \(v^{\prime }:=(r_z^d{\lambda })^\beta \exp _{d-1}^{-1}u:= (r_z^d{\lambda })^\beta v,\) where \(v \in \mathbb R ^{d-1}.\)

We put

$$\begin{aligned}&\displaystyle \mathcal T ^{{\lambda }, z}(K_z):= {K^{{\lambda }, z}}; \mathcal T ^{{\lambda }, z}(B_{r_z}(z_0)):= B^{{\lambda }, z};\\&\displaystyle \mathcal T ^{{\lambda }, z}(\mathcal{P }_{\lambda }\cap \mathcal K _z) := { \mathcal{P }^{{\lambda }, z}}; \ \ \ \mathcal T ^{{\lambda }, z}(\mathcal{P }_{\lambda }\cap B_{r_z}(z_0)) := { \mathcal{P }^{{\lambda }, z}_{r_z} }. \end{aligned}$$

We also have the a.e. equality \(B^{{\lambda }, z}=(r_z^d{\lambda })^\beta \mathbb{B _{d-1}}(\pi ) \times [0,(r_z^d{\lambda })^{2\beta })\), where \(\mathbb{B _{d-1}}(\pi )\) is the closure of the injectivity region of \(\exp _{d-1}\).

We next use the scaling transformations \(\mathcal T ^{{\lambda }, z}\) on \(\mathcal{A }_z(K)\) to define re-scaled \(k\)-face functionals \(\xi ^{{\lambda }, z}\) on re-scaled point sets \(\mathcal T ^{{\lambda }, z}(\mathcal{P }_{\lambda }\cap \mathcal K _z)\); in the sequel we show that these re-scaled functionals converge to the scaling limit functional \(\xi ^{(\infty )}\) given in Definition 2.2. In the special case that \(K\) is a ball, we remark that \(\mathcal{A }_z(\mathcal K )= K\) for all \(z \in \partial K\) and that \(\mathcal T ^{{\lambda }, z}\) coincide for all \(z \in \partial K\), putting us in the set-up of [8].

3.3 Re-scaled \(k\)-face functionals \(\xi ^{{\lambda }, z}, z \in \partial K, {\lambda }\ge 1\)

Fix \(K \in \mathcal{K }^3_+\), \({\lambda }\in [1, \infty )\) and \(z \in { \partial K }\). Let \(\xi _k, k \in \{0,1,\ldots , d-1\},\) be a generic \(k\) -face functional, as in Definition 2.1. The inverse transformation \([\mathcal T ^{{\lambda }, z}]^{-1}\) defines generic re-scaled \(k\)-face functionals \({\xi ^{{\lambda }, z} }({w^{\prime }}, \mathcal{X })\) defined for \({w^{\prime }}\in {K^{{\lambda }, z}}\) and \(\mathcal{X }\subset \mathbb R ^d\) by

$$\begin{aligned} {\xi ^{{\lambda }, z} }({w^{\prime }}, \mathcal{X }):= {\xi ^{{\lambda }, z}_k }({w^{\prime }}, \mathcal{X }):= \xi _k([\mathcal T ^{{\lambda }, z}]^{-1}({w^{\prime }}), [\mathcal T ^{{\lambda }, z}]^{-1}(\mathcal{X }\cap {K^{{\lambda }, z}})). \end{aligned}$$
(3.3)

It follows for all \(k \in \{0,1,\ldots ,d-1\}\), \(z \in { \partial K }, {\lambda }\in [1, \infty ),\) and \(x \in K_z\) that \(\xi _k(x,\mathcal{P }_{\lambda }\cap K_z):= {\xi ^{{\lambda }, z}_k }(\mathcal T ^{{\lambda }, z}(x), \mathcal P ^{{\lambda },z} ).\)

We shall establish properties of the re-scaled \(k\)-face functionals in the next section. For now, we record the distributional limit of the re-scaled point processes \({ \mathcal{P }^{{\lambda }, z}_{r_z} }\) as \({\lambda }\rightarrow \infty \).

Lemma 3.1

Fix \(z \in { \partial K }\). As \({\lambda }\rightarrow \infty \), there is a rate one homogeneous Poisson point process \(\tilde{\mathcal{P }}\) such that \({ \mathcal{P }^{{\lambda }, z}_{r_z} }\stackrel{\mathcal{D }}{\longrightarrow }\tilde{\mathcal{P }}\) in the sense of total variation convergence on compact sets.

Proof

This proof is a consequence of the discussion around (2.14) of [8], but for the sake of completeness we include the details. We find the image by \(\mathcal T ^{{\lambda }, z}\) of the measure on \(B_{r_z}(z_0)\) given by \({\lambda }r^{d-1} dr d\sigma _{d-1}(u)\). Under \(\mathcal T ^{{\lambda }, z}\) we have \(h^{\prime }:= (r_z^d{\lambda })^{2\beta }(1 - {r \over r_z})\), whence \(r = r_z(1 - (r_z^d{\lambda })^{-2\beta }h^{\prime }).\) Likewise we have \(v^{\prime }:=(r_z^d{\lambda })^{\beta } v\), whence \(v = (r_z^d{\lambda })^{-\beta } v^{\prime }\). Under \(\mathcal T ^{{\lambda }, z}\), the measure \(r^{d-1} dr\) becomes

$$\begin{aligned} r^{d-1} dr = (r_z(1 - (r_z^d{\lambda })^{-2 \beta }))^{d-1}r_z^{1 - 2 \beta d} {\lambda }^{-2 \beta } dh^{\prime } \end{aligned}$$

and \(d\sigma _{d-1}(u)\) transforms to

$$\begin{aligned} d\sigma _{d-1}(u) = \frac{ \sin ^{d-2}((r_z^d{\lambda })^{-\beta } |v^{\prime }|) }{ |(r_z^d{\lambda })^{-\beta } v^{\prime } |^{d-2} } (r_z^d{\lambda })^{- 1 + 2 \beta } dv^{\prime } \end{aligned}$$

as in (2.17) of [8]. Therefore the product measure \({\lambda }r^{d-1} dr d\sigma _{d-1}(u)\) transforms to

$$\begin{aligned} (1 - (r_z^d{\lambda })^{-2 \beta }h^{\prime })^{d-1} \frac{ \sin ^{d-2}({\lambda }^{-\beta } |v^{\prime }|) }{ |{\lambda }^{-\beta } v^{\prime } |^{d-2} } dh^{\prime } dv^{\prime }. \end{aligned}$$
(3.4)

The total variation distance between Poisson measures is upper bounded by a multiple of the \(L^1\) distance between their densities (Theorem 3.2.2 in [12]) and since \((1 - (r_z^d{\lambda })^{-2\beta })^{(d-1)} \rightarrow 1\) as \({\lambda }\rightarrow \infty \), the result follows. \(\square \)

4 Properties of the re-scaled \(k\)-face functional \({\xi ^{{\lambda }, z} }\)

Given \(k \in \{0,1,\ldots ,d-1\}, {\lambda }\ge 1, z \in { \partial K }\), we shall often write \(\xi \) for \({\xi ^{{\lambda }, z}_k }\) as at (3.3).

4.1 Localization of \(\xi ^{{\lambda }, z}\)

Fix \(K \in \mathcal{K }^3_+\). We appeal to results of Reitzner [14] to show that the re-scaled functionals \(\xi ^{{\lambda },z}\) ‘localize’, that is they are with high probability determined by ‘nearby’ point configurations.

For all \(s > 0\) consider the inner parallel set of \({ \partial K }\), namely

$$\begin{aligned} K(s):= \{x \in K \ : \ {\delta }^{H}(x, { \partial K }) \le s\}, \end{aligned}$$
(4.1)

with \({\delta }^H\) being the Hausdorff distance. Put

$$\begin{aligned} {\epsilon _{\lambda }}: = ( \frac{12 d \log {\lambda }}{ d_3 {\lambda }} )^{\beta }, \end{aligned}$$
(4.2)

where \(d_3:=d_3(K)\) is as in Lemma 5 of Reitzner [14]. We begin with two localization properties of the score \(\xi \).

Lemma 4.1

Fix \(K \in \mathcal{K }^3_+\) and \(k \in \{0,1,\ldots ,d-1\}\). (a) With probability at least \(1 - O({\lambda }^{-4d})\), for all \(z \in { \partial K }\), \(\rho \ge 1\), we have

$$\begin{aligned} \xi _k(x, \mathcal{P }_{\lambda }\cap K_z) = \left\{ \begin{array}{ll} \xi _k(x, \mathcal{P }_{\lambda }\cap K_z(\rho {\epsilon ^2_{\lambda }}))&\quad \mathrm{if} \ x \in K_z({\epsilon ^2_{\lambda }})\\ 0&\quad \mathrm{if} \ x \in K_z \setminus K_z( {\epsilon ^2_{\lambda }}). \end{array} \right. \end{aligned}$$
(4.3)

(b) There is a constant \(D_1\) such that for all \(z \in { \partial K }\) and \(x \in K_z({\epsilon ^2_{\lambda }})\) we have

$$\begin{aligned} P[\xi _k(x, \mathcal{P }_{\lambda }\cap K_z ) \ne \xi _k(x, \mathcal{P }_{\lambda }\cap K_z \cap B_{D_1 {\epsilon _{\lambda }}}(x))] = O({\lambda }^{-4d}). \end{aligned}$$

Proof

Throughout we shorthand \(\xi _k\) by \(\xi \). We prove part (a) with \(\rho = 1\). The proof for \(\rho > 1\) is identical. Let \(X_i, i \ge 1,\) be i.i.d. uniform on \(K_z\). For every integer \(l\), let \(A_l\) be the event that the boundary of co\((X_1,\ldots ,X_l)\) is contained in \(K_z (\epsilon _l^2)\).

Following nearly verbatim the discussion on p. 492 of [14], we note that \(P[A_l^c]\) equals the probability that at least one facet of \(\text{ co}(X_1,\ldots ,X_l)\) contains a point distant at least \(\epsilon _l^2\) from the boundary of \(K_z\), i.e., this is the probability that the hyperplane which is the affine hull of this facet cuts off from \(K_z\) a cap of height \(\epsilon _l^2\) which contains no point from \(X_1,\ldots ,X_l\). By Lemma 5 of [14], the volume of this cap is bounded by \(d_3 \epsilon _l^{d + 1} = 12d \log l/l\).

Thus when \(l\) is large enough so that \((l-d)/l > 1/2\) (ie. \(l > 2d\)) and \({(12 d \log l)/ l} < 1\), and using \(\log (1 -x) < -x, \ 0\! < x < \!1\), we get

$$\begin{aligned} P[A_l^c] \le \genfrac(){0.0pt}{}{l}{d} \left( 1 - {12 d \log l \over l} \right)^{l - d}&< l^d {1 \over d!} \exp \left( (l-d) ( - {12 d \log l \over l}) \right)\nonumber \\&\le {l^d \over d!} l^{-6d}= {l^{-5d}\over d!}. \end{aligned}$$
(4.4)

Let \(A_{\lambda }\) be the event that the boundary of \(\text{ co}(\mathcal{P }_{\lambda }\cap K)\) is contained in \(K_z(\epsilon _{\lambda }^2)\). Letting \(N({\lambda })\) be a Poisson random variable with parameter \({\lambda }\) we compute

$$\begin{aligned} P[A_{\lambda }^c] = \sum _{l= 0}^{\infty } P[A_l^c, N({\lambda }) = l] < \sum _{| l - {\lambda }| \le {\lambda }^{3/4}} P[A_l^c] + P[|N({\lambda }) - {\lambda }| \ge {\lambda }^{3/4}]. \end{aligned}$$

The last term decays exponentially with \({\lambda }\) and so exhibits growth \(O({\lambda }^{-4d})\). By (4.4), the first term has the same growth bounds since

$$\begin{aligned} \sum _{| l - {\lambda }| \le {\lambda }^{3/4}} P[A_l^c] \le 2 {\lambda }^{3/4} \max _{ |{\lambda }- l| \le {\lambda }^{3/4} }P[A_l^c] \le 2 {\lambda }^{3/4} {1 \over d!} ({\lambda }- {\lambda }^{3/4})^{-5d} = O({\lambda }^{-4d}), \end{aligned}$$

concluding the proof of (a).

We prove assertion (b). By part (a), it suffices to show there is \(\rho _0 \ge 1\) such that for \(x \in K_z({\epsilon ^2_{\lambda }})\)

$$\begin{aligned} P[\xi (x, \mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}) ) \ne \xi (x, \mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}) \cap B_{D_1 {\epsilon _{\lambda }}}(x))] = O({\lambda }^{-4d}). \end{aligned}$$

We consider the localization results described on pages 499–502 of [14] and in the Appendix of [14]. Using the set-up of Lemma 6 of [14], we choose \(m := m({\lambda }):= \lfloor (d_6 {\lambda }/(4d+1) \log {\lambda })^{(d-1)\beta } \rfloor \) points \(y_1,\ldots ,y_m\) on \({ \partial K }_z\) (here \(d_6:=d_6(K)\) is the constant of [14]) such that the Voronoi cells \(C_{{\mathrm{Vor}}}(y_j), 1 \le j \le m,\) partition \(K_z\), and such that the diameter of \(C_{{\mathrm{Vor}}}(y_j) \cap { \partial K }_z\) is \(O( {\epsilon _{\lambda }})\). Moreover, because all \(y_j\) are on \({ \partial K }_z\), any bisecting hyperplane between two \(y_j\) makes an angle with \(\partial K_z\) which is bounded from below. Consequently, since the ‘width’ of \(K_z( {\epsilon ^2_{\lambda }})\) is \(O({\epsilon ^2_{\lambda }})\), it follows that the diameter of the truncated cells \(C_{{\mathrm{Vor}}}(y_j) \cap K_z({\epsilon ^2_{\lambda }})\) is also \(O( {\epsilon _{\lambda }})\). Choose \(\rho _0\) large enough so that \(K_z(\rho _0 {\epsilon ^2_{\lambda }})\) contains the caps \(C_j, 1 \le j \le m,\) given near the end of p. 498 of [14].

For all \(1 \le j \le m\), let

$$\begin{aligned}S_j := \{ k \in \{1,2,\ldots ,m\}: \ C_{{\mathrm{Vor}}}(y_k) \cap C(y_j, d_{10} m^{-2\beta } ) \ne \emptyset \} \end{aligned}$$

where \(C(y, h)\) denotes a cap at \(y\) of height \(h\), and where \(d_{10}:=d_{10}(K)\) denotes the constant in [14]. Pages 498–500 of [14] show the existence of a set \(A^m\) such that \(P[A^m] \ge 1 - c_{16} {\lambda }^{-4d}\), \(c_{16}:=c_{16}(A)\), and on \(A^m\) the score \(\xi (x, \mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}))\) at \(x \in K_z({\epsilon ^2_{\lambda }}) \cap C_{{\mathrm{Vor}}}(y_j)\) is determined by the Poisson points belonging to

$$\begin{aligned} U_j := U_j(x):= \bigcup _{k \in S_j} C_\mathrm{Vor}(y_k) \cap K_z( {\epsilon ^2_{\lambda }}), \end{aligned}$$
(4.5)

where \(j:= j(x) \in \{1,\ldots ,m\}\) is such that \(C_{{\mathrm{Vor}}}(y_j)\) contains \(x\). (Actually [14] shows this for the score \(\xi (x, \mathcal{P }_{\lambda }\cap K_z)\) and not for \(\xi (x, \mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}) )\), but the proof is the same, since \(\rho _0\) is chosen so that \(K_z(\rho _0 {\epsilon ^2_{\lambda }})\) contains the caps \(C_j, 1 \le j \le m.\)) By Lemma 7 of [14], the cardinality of \(S_j\) is at most \(d_8( d_{10}^{1/2} m^{-\beta } m^\beta + 1)^{d + 1} = O(1)\), uniformly in \(1 \le j \le m\). This implies that on \(A^m\), the score \(\xi (x, \mathcal{P }_{\lambda }\cap K_z(\rho {\epsilon ^2_{\lambda }}))\) at \(x \in K_z({\epsilon ^2_{\lambda }}) \cap C_{{\mathrm{Vor}}}(y_j)\) is determined by the Poisson points in \(U_j\), whose diameter is bounded by a constant multiple of the diameter of the truncated cells \(C_{{\mathrm{Vor}}}(y_k) \cap K_z(\rho {\epsilon ^2_{\lambda }}), k \in S_j\), and is thus determined by points distant at most \(D_1 {\epsilon _{\lambda }}\) from \(x\), \(D_1\) a constant. Since \(P[A_m^c] \le c_{16} {\lambda }^{-4d}\), this proves assertion (b).\(\square \)

The next lemma shows localization properties of \({\xi ^{{\lambda }, z} }\). We first require more terminology.

Definition 4.1

For all \(z \in { \partial K }\),we put

$$\begin{aligned} S^{{\lambda },z}:= \mathcal T ^{{\lambda },z}( K_z({\epsilon ^2_{\lambda }}) \cap B_{2D_1 {\epsilon _{\lambda }}}(z)). \end{aligned}$$

Note that if \({w^{\prime }}\!=\! (v^{\prime },h^{\prime }) \!\in \! S^{{\lambda },z}\), then \(|v^{\prime }| \!\le \! D_2 (\log {\lambda })^\beta \) for some \(D_2\) not depending on \(z\) (here we use \(\sup _{z \in \partial K} r_z \le C\)). Also, define \(D_3\) by the relation \(2 [\sup _{z \in \partial K} r_z^{d \beta } ] D_1 {\lambda }^\beta {\epsilon _{\lambda }}= D_3(\log {\lambda })^\beta \). For all \(L > 0\) and \(v \in \mathbb R ^{d-1}\), denote by \(\mathcal C _L(v)\) the cylinder \( \{(v^{\prime },h) \in \mathbb R ^{d-1} \times \mathbb R : \ |v^{\prime } - v| \le L \}\). Due to the non-linearity of \(\mathcal T ^{{\lambda }, z}\), localization properties for \(\xi \) do not in general imply localization properties for \(\xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\). However, the next lemma says that if the inverse image of \({w^{\prime }}\) is close to \(z\), then \(\xi ({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\) suitably localizes.

Lemma 4.2

Fix \(K \in \mathcal{K }^3_+\) and \(k \in \{0,1,\ldots ,d-1\}\). Uniformly in \(z \in { \partial K }\) and \({w^{\prime }}:=(v^{\prime },h^{\prime }) \in S^{{\lambda },z}\) we have

$$\begin{aligned} P[ {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) \ne {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cap \mathcal C _{D_3 (\log {\lambda })^\beta } (v^{\prime } )) ] = O ({\lambda }^{-4d}). \end{aligned}$$

Remark

When \(K\) is the unit ball we show in [8] that the scores \(\xi \) localize in the following stronger sense: for all \({w^{\prime }}:= (v^{\prime },h^{\prime }) \in \mathcal K ^{{\lambda }, z}\), there is an a.s. finite random variable \(R:= R({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\) such that

$$\begin{aligned} {\xi ^{{\lambda }, z} }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) = {\xi ^{{\lambda }, z} }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cap \mathcal C _{r} (v^{\prime } ) ) \end{aligned}$$
(4.6)

for all \(r \ge R\), with \(\sup _{{\lambda }} P[R>t] \rightarrow 0\) as \(t \rightarrow \infty \). We are unable to show this latter property for arbitrary smooth \(K\).

Proof

Fix the reference boundary point \(z \in { \partial K }\) and write \({\xi ^{{\lambda }, z} }\) for \({\xi ^{{\lambda }, z}_k }\). Let \(\rho _0\) be as in the proof of Lemma 4.1(b). For any \(A \subset \mathbb R ^{+} \times \mathbb R ^{d-1}\), we let \(\mathcal T ^{{\lambda },z}(A):= A^{{\lambda }, z}.\) In view of Lemma 4.1(b), it suffices to show for \({w^{\prime }}:=(v^{\prime },h^{\prime }) \in S^{{\lambda },z}\) that

$$\begin{aligned} P[ {\xi ^{{\lambda }, z} }({w^{\prime }}, (\mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}))^{{\lambda }, z} )&\ne {\xi ^{{\lambda }, z} }({w^{\prime }}, (\mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}))^{{\lambda }, z} \cap \mathcal C _{D_3 (\log {\lambda })^\beta } (v^{\prime } )) ]\\&= O({\lambda }^{-4d}). \end{aligned}$$

Given \(w^{\prime }\), find \(j:=j(w^{\prime })\) such that \( C_{{\mathrm{Vor}}}(y_j)\) contains \([\mathcal T ^{{\lambda },z}]^{-1}(w^{\prime }):=x\). Recall the definition of \(U_j:= U_j(x)\) at (4.5) and recall that the proof of Lemma 4.1 shows that \(\text{ diam}(U_j) \le D_1 \epsilon _{\lambda }\). By the \(C^3\) assumption, if \({\lambda }\) is large then for all \(z \in { \partial K }\) the projection of \(U_j\) onto the osculating sphere at \(z\) has a diameter comparable to that of \(U_j\), i.e., is generously bounded by \(2D_1 \epsilon ({\lambda })\). Thus the spatial diameter of \(\mathcal T ^{{\lambda },z}(U_j)\) is bounded by \( 2 [\sup _{z \in \partial K} r_z^{d \beta } ] {\lambda }^{\beta } D_1 {\epsilon _{\lambda }}= D_3(\log {\lambda })^{\beta }\), by definition of \(D_3\). In other words

$$\begin{aligned} \mathcal T ^{{\lambda },z}(U_j) \subset \mathcal C _{D_3 (\log {\lambda })^\beta } (v^{\prime }). \end{aligned}$$
(4.7)

However, as seen in the proof of Lemma 4.1, with probability at least \(1 - c_{16} {\lambda }^{-4d}\), the score \({\xi ^{{\lambda }, z} }({w^{\prime }}, (\mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}))^{{\lambda }, z} )\) is determined by the points \((\mathcal{P }_{\lambda }\cap K_z(\rho _0 {\epsilon ^2_{\lambda }}))^{{\lambda }, z}\) in \(\mathcal T ^{{\lambda },z}(U_j)\). In view of (4.7), the proof is complete. \(\square \)

4.2 Moment bounds for \({\xi ^{{\lambda }, z} }\)

We use the localization results to derive moment bounds for the re-scaled \(k\)-face functionals \({\xi ^{{\lambda }, z} }\). Given a random variable \(W\) and \(p > 0\), we let \(||W||_p:= (\mathbb E |W|^p)^{1/p}\).

Lemma 4.3

Fix \(K \in \mathcal{K }^3_+\) and \(k \in \{0,1,\ldots ,d-1\}.\) For all \(p \in [1, 4]\) there are constants \(M(p):= M(p,k) \in (0, \infty )\) such that

$$\begin{aligned} \sup _{z \in { \partial K }} \sup _{{\lambda }\ge 1} \sup _{{w^{\prime }}\in B^{{\lambda }, z} } || \xi ^{{\lambda },z}_k({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }) ||_p \le M(p) \end{aligned}$$
(4.8)

and

$$\begin{aligned} \sup _{z \in { \partial K }} \sup _{{\lambda }\ge 1} \sup _{{w^{\prime }}\in S^{{\lambda },z} } || \xi ^{{\lambda },z}_k({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) ||_p \le M(p) (\log {\lambda })^{k}. \end{aligned}$$
(4.9)

Proof

The bound (4.8) follows as in Lemma 7.1 of [8]. To prove (4.9), we argue as follows. Given \(z \in { \partial K }\) and \({w^{\prime }}\in S^{{\lambda },z} \), we let

$$\begin{aligned} E := E_z ({w^{\prime }}):= \left\{ {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) = {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cap \mathcal C _{D_3 (\log {\lambda })^\beta } (v^{\prime } ) \cap (K_z({\epsilon ^2_{\lambda }}))^{{\lambda }, z} ) \right\} \!. \end{aligned}$$

By Lemmas 4.1(a) and 4.2 we have \(P[E^c] = O({\lambda }^{-4d}).\)

Let \(N(s)\) be a Poisson random variable with parameter \(s\). The cardinality of the point set

$$\begin{aligned} { \mathcal{P }^{{\lambda }, z}}\cap \mathcal C _{D_3 (\log {\lambda })^\beta } (v^{\prime } ) \cap (K_z({\epsilon ^2_{\lambda }}))^{{\lambda }, z}, \end{aligned}$$

is stochastically bounded by \(N(C (\log {\lambda })^{{\beta }(d-1)} \cdot (\log {\lambda })^{2 \beta }) = N(C \log {\lambda })\), where \(\mathcal C \) is a generic constant whose value may change from line to line. On the event \(E\) the number of \(k\)-faces containing \({w^{\prime }}\) is generously bounded by \( \genfrac(){0.0pt}{}{ N(C \log {\lambda }) }{k} \le (N(C \log {\lambda }))^{k} \).

We now compute for \(p \in [1,4]\):

$$\begin{aligned} || \xi ^{{\lambda }, z}_k({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})||_p \le || \xi ^{{\lambda }, z}_k({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{{1}}(E)||_p + || \xi ^{{\lambda }, z}_k({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{{1}}(E^c)||_p. \end{aligned}$$

The first term is bounded by \((k+1)^{-1}||N^{k}(C \log {\lambda })||_p \le M(p) (\log {\lambda })^{k}\).

The second term is bounded by

$$\begin{aligned} \frac{1}{k+1} \left| \left| \genfrac(){0.0pt}{}{\text{ card}(\mathcal{P }_{\lambda })}{k} \right| \right|_{pr} {\lambda }^{-4d/pq}, \ \ 1/r + 1/q = 1. \end{aligned}$$

We have \(|| \genfrac(){0.0pt}{}{\text{ card}(\mathcal{P }_{\lambda })}{k} ||_{pr} = O({\lambda }^{k})\) and for \(p \in [1,4]\) we may choose \(q\) sufficiently close to \(1\) such that \({\lambda }^{-4d/pq} = O({\lambda }^{-k})\). This gives (4.9).

Remark

  1. (i)

    Straightforward modifications of the proof of Lemma 4.1 show that the \(O({\lambda }^{-4d})\) bounds of that lemma may be replaced by \(O({\lambda }^{-md})\) bounds, \(m\) an arbitrary integer, provided that \(\epsilon _{\lambda }\) given at (4.2) is increased by a scalar multiple of \(m\). In this way one could show that Lemma 4.3 holds for moments of any order \(p > 0\). Since we do not require more than fourth moments for \(\xi ^{{\lambda },z}\), we do not strive for this generality.

  1. (ii)

    We do not claim that the bounds of Lemma 4.3 are optimal. By McMullen’s bound [10], the \(k\) face functional on an \(n\) point set is bounded by \(Cn^{d/2}\) and using this bound for \(k > d/2\) shows that the \((\log {\lambda })^k\) term in (4.9) can be improved to \((\log {\lambda })^{d/2}\). The \(\log {\lambda }\) factors could possibly be dispensed with altogether, as mentioned in the remark at the end of Sect. 5.2.

4.3 Comparison of scores for points in a ball and on \(K_z\)

The \(k\)-face functional of Definition 2.1 on Poisson input on the ball is well understood [8]. To exploit this we need to show that the re-scaled functional \({\xi ^{{\lambda }, z} }\) on \({ \mathcal{P }^{{\lambda }, z}}\) is well approximated by its value on \({ \mathcal{P }^{{\lambda }, z}_{r_z} }\). We shall also need to show that the pair correlation function for \({\xi ^{{\lambda }, z} }\) on \({ \mathcal{P }^{{\lambda }, z}}\) is well approximated by the pair correlation function for \({\xi ^{{\lambda }, z} }\) on \({ \mathcal{P }^{{\lambda }, z}_{r_z} }\). These approximations are established in the next four lemmas.

Our first lemma records a simple geometric fact. Locally around \(z\), the osculating ball to \(K_z\) may lie inside or outside \(K_z\), but it is not far from \({ \partial K }_z\). The next lemma shows that the distance decays like the cube of \(|v^{\prime }|\).

Lemma 4.4

Fix \(K \in \mathcal{K }^3_+\). For all \(z \in { \partial K }\) and \(v:= (r_z^d {\lambda })^{-\beta } v^{\prime }\) we have

$$\begin{aligned} r_z^{2\beta d} {\lambda }^{2 \beta }\left|1 - {f(\exp _{d-1} (v) ) \over r_z}\right| \le D_4 r_z^{-1 - \beta d} {\lambda }^{-\beta } |v^{\prime }|^3. \end{aligned}$$
(4.10)

Proof

We first show (4.10) when \(d=2\). The boundary of the osculating circle at \(z\) coincides with \({ \partial K }\) up to at least second order, giving \(f(0) = r_z, f^{\prime }(0) = f^{\prime \prime }(0) = 0\). The Taylor expansion for \(f\) around \(0\) gives \(|1 - {f(v) \over r_z}| \le {1 \over 6} ||f^{^{\prime \prime \prime }}||_{\infty } r_z^{-1} |v|^3\), whence the result.

Consider the case \(d \ge 3\). Let \(\exp _{d-1} (v) := \cos (|v|) k_z + \sin (|v|) w\), where \(w := v/|v|\). It is enough to consider the section of the osculating ball and \(K_z\) with the plane generated by \(k_z\) and \(w\). Indeed, we obtain in that plane a two-dimensional mother body with an osculating radius equal to \(r_z\) at the point \(z\). We may apply the case \(d=2\) to deduce the required result. \(\square \)

Lemma 4.5

Fix \(K \in \mathcal{K }^3_+\) and \(k \in \{0,1,\ldots ,d-1\}\). Uniformly for \(z \in { \partial K }\) and \({w^{\prime }}\in S^{{\lambda },z} \cap B^{{\lambda }, z}\), we have

$$\begin{aligned} \mathbb E \left| {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) - {\xi ^{{\lambda }, z}_k }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }) \right| = O\left( {\lambda }^{-{\beta }/2} (\log {\lambda })^{k + ({\beta }+ 1)/2} \right)\!. \end{aligned}$$
(4.11)

Proof

Write \({\xi ^{{\lambda }, z} }\) for \({\xi ^{{\lambda }, z}_k }\). For \({w^{\prime }}\in S^{{\lambda },z} \cap B^{{\lambda }, z}\), we put

$$\begin{aligned} E&:= E({w^{\prime }}) := \{{\xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) = \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cap \mathcal C _{D_3 (\log {\lambda })^\beta } ({w^{\prime }}))}\nonumber \\&\cup \,\{\xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }) = \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cap \mathcal C _{D_3 (\log {\lambda })^\beta } ({w^{\prime }}))\}, \end{aligned}$$
(4.12)

so that \(P[E^c] = O({\lambda }^{-4d})\) by Lemma 4.2. Put

$$\begin{aligned} F^{{\lambda },z}({w^{\prime }}):= \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) - \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }). \end{aligned}$$

By Lemma 4.3 with \(p = 2\), we have \(||F^{{\lambda },z}({w^{\prime }})||_2 \le 2 M(2) (\log {\lambda })^{k},\) uniformly in \({w^{\prime }}, {\lambda }\) and \(z\).

Recall \({w^{\prime }}:= (v^{\prime },h^{\prime })\). For all \({w^{\prime }}\in S^{{\lambda },z} \cap B^{{\lambda }, z}\) put

$$\begin{aligned} R({w^{\prime }}):=&\{(v^{\prime \prime },h^{\prime \prime }) \in \mathbb R ^{d-1} \times \mathbb R : \ |v^{\prime \prime } - v^{\prime }| \le D_3 (\log {\lambda })^\beta ,\nonumber \\&|h^{\prime \prime }| \le (r_z^d {\lambda })^{2\beta }|1 - r_z^{-1}f( \exp _{d-1}((r_z^d {\lambda })^{-\beta } v^{\prime \prime }) ) |\}. \end{aligned}$$
(4.13)

Write

$$\begin{aligned} \mathbb E |F^{{\lambda },z}({w^{\prime }})| = \mathbb E |(F^{{\lambda },z}({w^{\prime }}))(\mathbf{1 }(E) + \mathbf{1 }(E^c))| .\end{aligned}$$

On \(E\) we have \(F^{{\lambda },z}({w^{\prime }})= 0\), unless the realization of \({ \mathcal{P }^{{\lambda }, z}}\) puts points in the set \(R({w^{\prime }})\). By the Cauchy–Schwarz inequality and Lemma 4.3 with \(p = 2\) there, we have

$$\begin{aligned} \mathbb E |( F^{{\lambda },z}({w^{\prime }}))\mathbf{1 }(E)| \le 2M(2) (\log {\lambda })^k (P[ \mathbf{1 }({ \mathcal{P }^{{\lambda }, z}}\cap R(v^{\prime }) \ne \emptyset )])^{1/2}. \end{aligned}$$
(4.14)

The Lebesgue measure of \(R({w^{\prime }})\) is bounded by the product of the area of its ‘base’, that is \((2D_3 (\log {\lambda })^\beta )^{d-1}\) and its ‘height’, which by Lemma 4.4 is at most \(D_4 r_z^{-1- \beta d}{\lambda }^{-\beta } \left(|v^{\prime }| + D_3 (\log {\lambda })^\beta \right)^3.\)

By (3.4), the \({ \mathcal{P }^{{\lambda }, z}}\) intensity measure of \(R({w^{\prime }})\), denoted by \(|R({w^{\prime }})|\), thus satisfies

$$\begin{aligned} |R({w^{\prime }})| \le (2D_3 (\log {\lambda })^\beta )^{d-1}D_4 r_z^{-1- \beta d}{\lambda }^{-\beta } (|v^{\prime }| + D_3 (\log {\lambda })^\beta )^3. \end{aligned}$$
(4.15)

Since \(1 - e^{-x} \le x\) holds for all \(x\) it follows that

$$\begin{aligned} P[\mathbf{1 }({ \mathcal{P }^{{\lambda }, z}}\cap R(v^{\prime }) \ne \emptyset )] = 1 - \exp (-|R({w^{\prime }})| ) \le |R({w^{\prime }})|. \end{aligned}$$
(4.16)

Combining (4.14)–(4.16), and recalling that \(|v^{\prime }| \le D_2 (\log {\lambda })^\beta \), shows that \(\mathbb E |(F^{{\lambda },z}({w^{\prime }}))\mathbf{1 }(E)|\) is bounded by the right hand side of (4.11).

Similarly, Lemma 4.3, the bound \(P[E^c] = O( {\lambda }^{-4d})\), and the Cauchy–Schwarz inequality give \(\mathbb E |(F^{{\lambda },z}({w^{\prime }}))\mathbf{1 }(E^c)| = O((\log {\lambda })^k {\lambda }^{-2d})\), which is dominated by the right hand side of (4.11). Thus (4.11) holds as claimed. \(\square \)

The next two lemmas justify identifying \(\xi ^{(\infty )}\) as a scaling limit, as given by Definition 2.2. The first lemma is a restatement of Lemma 7.2 in [8].

Lemma 4.6

For all \(k \in \{0,1,\ldots ,d-1\}\), \(z \in { \partial K }\) and \(h\ge 0\) we have

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty }| \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}_{r_z} }) - \mathbb E \xi ^{(\infty )}_k((\mathbf{0},h), \mathcal{P })| = 0. \end{aligned}$$

Lemma 4.7

Fix \(K \in \mathcal{K }^3_+\). For all \(k \in \{0,1,\ldots ,d-1\}\), \(z \in { \partial K }\) and \((\mathbf{0},h) \in {K^{{\lambda }, z}}\) we have

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty }| \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}}) - \mathbb E \xi ^{(\infty )}_k((\mathbf{0},h), \mathcal{P }) |= 0. \end{aligned}$$

Proof

We bound \(| \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}}) - \mathbb E \xi ^{(\infty )}_k((\mathbf{0},h), \mathcal{P }) |\) by

$$\begin{aligned} | \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}}) \!-\! \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}_{r_z} }) | \!+\! | \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}_{r_z} }) \!-\! \mathbb E \xi ^{(\infty )}_k((\mathbf{0},h), \mathcal{P })|. \end{aligned}$$

The first term goes to zero by Lemma 4.5 with \(w^{\prime } = (\mathbf{0},h)\) and the second term goes to zero by Lemma 4.6. \(\square \)

We next recall the definition of the pair correlation function for the score \(\xi \) as well as for its re-scaled version.

Definition 4.2

(Pair correlation functions) For all \(x, y \in K_z\), any random point set \(\Xi \subset K_z\), and any \(\xi \) we put

$$\begin{aligned} c(x,y; \Xi ) := c^\xi (x,y; \Xi ):= \mathbb E \xi (x, \Xi \cup y) \xi (y, \Xi \cup x) - \mathbb E \xi (x, \Xi ) \mathbb E \xi (y, \Xi ).\nonumber \\ \end{aligned}$$
(4.17)

For all \(k \in \{0,1,\ldots ,d-1\}\), \({\lambda }\ge 1, z \in { \partial K }, \ (\mathbf{0}, h) \in {K^{{\lambda }, z}},\) and \((v^{\prime },h^{\prime }) \in {K^{{\lambda }, z}}\), define the re-scaled pair correlation function for \({\xi ^{{\lambda }, z}_k }\) with respect to \({ \mathcal{P }^{{\lambda }, z}}\) by

$$\begin{aligned}&{ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\nonumber \\&\quad := c^{{\lambda }, z}_k((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\nonumber \\&\quad \ =\mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}}\cup (v^{\prime },h^{\prime })) {\xi ^{{\lambda }, z}_k }((v^{\prime },h^{\prime }), { \mathcal{P }^{{\lambda }, z}}\cup (\mathbf{0},h)) \nonumber \\&\qquad \ \,- \mathbb E {\xi ^{{\lambda }, z}_k }((\mathbf{0},h), { \mathcal{P }^{{\lambda }, z}}) \mathbb E {\xi ^{{\lambda }, z}_k }((v^{\prime },h^{\prime }), { \mathcal{P }^{{\lambda }, z}}). \end{aligned}$$
(4.18)

The next lemma shows that the pair correlation function for \(\xi ^{{\lambda },z}\) on \({ \mathcal{P }^{{\lambda }, z}}\) is well approximated by the pair correlation function for \(\xi ^{{\lambda },z}\) on \({ \mathcal{P }^{{\lambda }, z}_{r_z} }\).

Lemma 4.8

Fix \(K \in \mathcal{K }^3_+\) and \(k \in \{0,1,\ldots ,d-1\}\). Uniformly for \(z \in { \partial K }\), \({{w_0}^{\prime }}:= (\mathbf{0}, h) \in S^{{\lambda },z} \cap B^{{\lambda }, z}\) and \( {w^{\prime }}:= (v^{\prime },h^{\prime }) \in S^{{\lambda },z} \cap B^{{\lambda }, z}\), we have

$$\begin{aligned} |c^{{\lambda }, z}_k({{w_0}^{\prime }}, {w^{\prime }}; { \mathcal{P }^{{\lambda }, z}}) - c^{{\lambda }, z}_k({{w_0}^{\prime }}, {w^{\prime }}; { \mathcal{P }^{{\lambda }, z}_{r_z} })| = O \left( {\lambda }^{-{\beta }/3}(\log {\lambda })^{2k + ({\beta }+ 1)/3} \right)\!.\qquad \end{aligned}$$
(4.19)

Proof

It suffices to modify the proof of Lemma 4.5. Write \(c^{{\lambda },z}\) and \(\xi ^{{\lambda },z}\) for \(c^{{\lambda }, z}_k\) and \({\xi ^{{\lambda }, z}_k }\), respectively. Put \(F:= E({{w_0}^{\prime }}) \cap E({w^{\prime }})\), where \(E({{w_0}^{\prime }})\) and \(E({w^{\prime }})\) are defined at (4.12). We have \(P[F^c] = O( {\lambda }^{-4d})\) by Lemma 4.2. Write

$$\begin{aligned}&\mathbb E \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup (v^{\prime },h^{\prime })) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup {{w_0}^{\prime }}) \nonumber \\&\qquad \quad - \mathbb E \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {w^{\prime }}) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {{w_0}^{\prime }}) \nonumber \\&\qquad = \mathbb E [ \{\xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup {w^{\prime }}) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup {{w_0}^{\prime }}) \nonumber \\&\quad \quad \quad - \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {w^{\prime }}) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {{w_0}^{\prime }})\} \mathbf{{1}} (F)] \nonumber \\&\qquad \quad + \ \mathbb E [\{\xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup {w^{\prime }}) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}\cup {{w_0}^{\prime }}) \nonumber \\&\quad \quad \quad - \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {w^{\prime }}) \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }\cup {{w_0}^{\prime }})\} \mathbf{{1}}(F^c)]\nonumber \\&\qquad := I_1 + I_2. \end{aligned}$$
(4.20)

The random variable in the expectation \(I_1\) vanishes, except on the event

$$\begin{aligned} H({{w_0}^{\prime }},{w^{\prime }}):= \{ { \mathcal{P }^{{\lambda }, z}}\cap R({w^{\prime }}) \ne \emptyset \} \cup \{ { \mathcal{P }^{{\lambda }, z}}\cap R({{w_0}^{\prime }}) \ne \emptyset \}, \end{aligned}$$

where \(R({w^{\prime }})\) and \(R({{w_0}^{\prime }})\) are at (4.13). The Hölder inequality \(||UVW||_1 \le ||U||_{3} ||V||_{3} ||W||_3\) for random variables \(U, V, W\) and Lemma 4.3 with \(p = 3\) imply that

$$\begin{aligned} I_1 \le 2 (M(3))^2 (\log {\lambda })^{2k} (P[H({{w_0}^{\prime }},{w^{\prime }})])^{1/3}, \end{aligned}$$

that is to say

$$\begin{aligned} I_1 = O \left( (\log {\lambda })^{2k} \left( r_z^{-1- {\beta }d}{\lambda }^{-{\beta }} (\log {\lambda })^{{\beta }(d-1)} [ (|v^{\prime }| + D_3 (\log {\lambda })^{\beta })^3 + (D_3 (\log {\lambda })^{\beta })^3] \right)^{1/3} \right)\!, \end{aligned}$$

which for \(|v^{\prime }| \le D_2(\log {\lambda })^{\beta }\) satisfies the growth bounds on the right hand side of (4.19).

Now term \(I_2\) in (4.20) is bounded by \(2(M(3))^2 (\log {\lambda })^{2k}(P[F^c])^{1/3}\), which is of smaller order than the right hand side of (4.19). This shows that (4.20) also satisfies the growth bounds on the right hand side of (4.19).

It remains to bound

$$\begin{aligned} |\mathbb E \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) \mathbb E \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}}) - \mathbb E \xi ^{{\lambda },z}({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} }) \mathbb E \xi ^{{\lambda },z}({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} })|. \end{aligned}$$
(4.21)

Notice that two applications of the Cauchy–Schwarz inequality give that the difference (4.21) differs from

$$\begin{aligned}&|\mathbb E [{\xi ^{{\lambda }, z} }({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{1 }(F)] \mathbb E [{\xi ^{{\lambda }, z} }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{1 }(F)] \nonumber \\&\quad - \mathbb E [{\xi ^{{\lambda }, z} }({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} })\mathbf{1 }(F)] \mathbb E [\xi ({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} })\mathbf{1 }(F)]| \end{aligned}$$
(4.22)

by at most

$$\begin{aligned} 4 M(1) M(2) (\log {\lambda })^{2k} P[F^c]^{1/2}= O( (\log {\lambda })^{2k} {\lambda }^{-2d}), \end{aligned}$$

which is of smaller order than the right hand side of (4.19).

Now we control the difference (4.22) which we write as \(|\mathbb E e_1 \mathbb E e_2 - \mathbb E e_3 \mathbb E e_4|\), where \(e_1 := {\xi ^{{\lambda }, z} }({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{1}(F)\), \(e_2:= {\xi ^{{\lambda }, z} }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}})\mathbf{1}(F)\), \(e_3:= {\xi ^{{\lambda }, z} }({{w_0}^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} })\mathbf{1}(F)\), and \(e_4:= {\xi ^{{\lambda }, z} }({w^{\prime }}, { \mathcal{P }^{{\lambda }, z}_{r_z} })\mathbf{1}(F)\). The proof of Lemma 4.5 (with \(E\) replaced by \(F\)) shows that

$$\begin{aligned} \mathbb E |e_1 - e_3| = O( {\lambda }^{-{\beta }/2 } (\log {\lambda })^{k + ({\beta }+ 1)/2}) \end{aligned}$$

and

$$\begin{aligned} \mathbb E |e_2 - e_4| = O( {\lambda }^{-{\beta }/2 } (\log {\lambda })^{k + ({\beta }+ 1)/2}). \end{aligned}$$

Since \( |\mathbb E e_1 \mathbb E e_2 - \mathbb E e_3 \mathbb E e_4| \le |\mathbb E e_1| |\mathbb E e_2 - \mathbb E e_4| + |\mathbb E e_4 | |\mathbb E e_1 - \mathbb E e_3| \) it follows that (4.21) is bounded by

$$\begin{aligned} O({\lambda }^{-{\beta }/2 } (\log {\lambda })^{2k + ({\beta }+ 1)/2}) + O( (\log {\lambda })^{2k} {\lambda }^{-4d/3}), \end{aligned}$$

i.e., is bounded by the right-hand side of (4.19).\(\square \)

Letting \(\xi :=\xi _k\) denote a generic \(k\)-face functional, our last lemma describes a decay rate for \(c^\xi (x,y; \mathcal{P }_{\lambda }\cap K_z)\) which, while not optimal, is enough for proving Theorem 2.1. \(\square \)

Lemma 4.9

Fix \(K \in \mathcal{K }^3_+\). For all \(z \in { \partial K }\) and \(x ,y \in K_z({\epsilon ^2_{\lambda }})\) with \(|x - y| \ge 2D_1 {\epsilon _{\lambda }}\), we have

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{ 1 + 2 {\beta }} c^\xi (x,y; \mathcal{P }_{\lambda }\cap K_z) =0. \end{aligned}$$

Proof.

Fix \(x \in K_z({\epsilon ^2_{\lambda }})\). To lighten the notation we abbreviate \(\mathcal{P }_{\lambda }\cap K_z\) by \(\mathcal{P }_{\lambda }\) in this proof only. For \(y \in K_z({\epsilon ^2_{\lambda }})\), put

$$\begin{aligned} E\!:=\! E(x,y)\!:=\! \{\xi (x, \mathcal{P }_{\lambda }) \!=\! \xi (x, \mathcal{P }_{\lambda }\!\cap \! B_{D_1 {\epsilon _{\lambda }}}(x)) \} \!\cup \! \{\xi (y, \mathcal{P }_{\lambda })\!=\! \xi (y, \mathcal{P }_{\lambda }\!\cap \! B_{D_1 {\epsilon _{\lambda }}}(y)) \}. \end{aligned}$$

Lemma 4.1(b) gives

$$\begin{aligned} P[E^c] = O( {\lambda }^{-4d}). \end{aligned}$$
(4.23)

If \(|x - y| \ge 2D_1 {\epsilon _{\lambda }}\), then \(\xi (x, \mathcal{P }_{\lambda }\cup y)\) and \(\xi (y, \mathcal{P }_{\lambda }\cup x)\) are independent on \(E\), giving

$$\begin{aligned} \mathbb E [\xi (x, \mathcal{P }_{\lambda }\cup y) \xi (y, \mathcal{P }_{\lambda }\cup x) \mathbf{1}(E)]&= \mathbb E [\xi (x, \mathcal{P }_{\lambda }) \mathbf{1}(E) \cdot \xi (y, \mathcal{P }_{\lambda }) \mathbf{1}(E)]\\&= \mathbb E [\xi (x, \mathcal{P }_{\lambda }) \mathbf{1}(E)] \cdot \mathbb E [ \xi (y,\mathcal{P }_{\lambda }) \mathbf{1}(E)]. \end{aligned}$$

Writing \(\mathbf{1}(E) = 1 - \mathbf{1}(E^c)\) gives

$$\begin{aligned}&\mathbb E [\xi (x, \mathcal{P }_{\lambda }\cup y) \xi (y, \mathcal{P }_{\lambda }\cup x) \mathbf{1}(E)]\\&\quad = \left(\mathbb E \xi (x, \mathcal{P }_{\lambda }) - \mathbb E [\xi (x, \mathcal{P }_{\lambda }) \mathbf{1}(E^c)]\right) \cdot \left(\mathbb E \xi (y, \mathcal{P }_{\lambda }) - \mathbb E [\xi (y,\mathcal{P }_{\lambda }) \mathbf{1}(E^c)]\right) \\&\quad = \mathbb E \xi (x, \mathcal{P }_{\lambda })\mathbb E \xi (y, \mathcal{P }_{\lambda }) +G(x,y), \end{aligned}$$

where

$$\begin{aligned} G(x,y)&:= - \mathbb E \xi (x, \mathcal{P }_{\lambda })\mathbb E [\xi (y, \mathcal{P }_{\lambda }) \mathbf{1}(E^c)]\\&- \ \mathbb E \xi (y, \mathcal{P }_{\lambda })\mathbb E [\xi (x, \mathcal{P }_{\lambda }) \mathbf{1}(E^c)] + \mathbb E [\xi (x,\mathcal{P }_{\lambda }) \mathbf{1}(E^c)] \cdot \mathbb E [\xi (y, \mathcal{P }_{\lambda }) \mathbf{1}(E^c)]. \end{aligned}$$

Let \(N({\lambda }) := \text{ card}(\mathcal{P }_{\lambda }\cap K_z)\). By McMullen’s bounds [10] for the number of \(k\)-dimensional faces and standard moment bounds for Poisson random variables we have \(||\xi (x, \mathcal{P }_{\lambda })||_1 \le C ||N^{d/2}({\lambda })||_1 \le C {\lambda }^{d/2}\) and similarly \( ||\xi (y, \mathcal{P }_{\lambda })||_2 \le C {\lambda }^{d/2}.\) By the Cauchy–Schwarz inequality, it follows that

$$\begin{aligned} |\mathbb E \xi (x, \mathcal{P }_{\lambda })\mathbb E [\xi (y, \mathcal{P }_{\lambda }) \mathbf{1}(E^c)]| = O({\lambda }^{d/2} {\lambda }^{d/2} (P[E^c])^{1/2}) = o({\lambda }^{-1 - 2{\beta }}), \end{aligned}$$

where the last estimate easily follows by (4.23). The other two terms comprising \(G(x,y)\) have the same asymptotic behavior and so \(G(x,y) = o({\lambda }^{-1 - 2{\beta }})\).

On the other hand, \(\mathbb E [\xi (x, \mathcal{P }_{\lambda }\cup y) \xi (y, \mathcal{P }_{\lambda }\cup x) \mathbf{1}(E)]\) differs from \(\mathbb E \xi (x, \mathcal{P }_{\lambda }\cup y)) \xi (y, \mathcal{P }_{\lambda }\cup x)\) by \(\mathbb E [\xi (x, \mathcal{P }_{\lambda }\cup y) \xi (y,\mathcal{P }_{\lambda }\cup x) \mathbf{1}(E^c)].\) The Hölder inequality \(||U V W||_1 \le ||U||_4 ||V||_4 ||W||_2\) shows that this term is \(o({\lambda }^{-1 - 2{\beta }})\).

Thus \(\mathbb E \xi (x, \mathcal{P }_{\lambda }\cup y)) \xi (y,\mathcal{P }_{\lambda }\cup x)\) and \(\mathbb E \xi (x, \mathcal{P }_{\lambda })\mathbb E \xi (y, \mathcal{P }_{\lambda })\) differ from \(\mathbb E [\xi (x, \mathcal{P }_{\lambda }\cup y) \xi (y,\mathcal{P }_{\lambda }\cup x) \mathbf{1}(E)]\) by \(o({\lambda }^{-1 - 2{\beta }})\), concluding the proof of Lemma 4.9. \(\square \)

5 Proof of Theorem 2.1

Fix \(K \in \mathcal{K }^3_+\). Recall that \(\mathcal{M }(K)\) denotes the medial axis of \(K\) and, for every \(z\in \partial K\) the inner unit-normal vector of \(\partial K\) at \(z\) is \(k_z\). Put \(t(z):=\inf \{t>0:z+ tk_z\in \mathcal{M }(K)\}\). In particular, for every \(t<t(z)\), the ball \(B_t(z+tk_z)\) is included in \(K\) so \(t(z)<\infty \) because of the boundedness of \(K\). Thus the map \(\varphi :(z,t) \longmapsto (z+tk_z)\) is a diffeomorphism from \(\{(z,t):z\in \partial K,0<t<t(z)\}\) to \(\text{ Int}(K)\setminus \mathcal{M }(K)\). In particular, \(z\longmapsto -k_z\) is the Gauss map and its ifferential is the shape operator or Weingarten map \(W_z\), which we recall has eigenvalues \(C_{z,1},\cdots ,C_{z,d-1}\). Consequently, the Jacobian of \(\varphi \) may be written as \(\det (I-tW_z)=\prod _{i=1}^{d-1}(1-tC_{z,i}).\)

5.1 Proof of expectation asymptotics (2.4)

With \(K \in \mathcal{K }^3_+\) fixed, fix \(g \in \mathcal{C }(K)\) and let \(\xi \) and \(\mu _{\lambda }^\xi \) denote a generic \(k\) face functional and \(k\) face measure, respectively. Recall that we may uniquely write \(x \in K {\setminus } \mathcal{M }(K)\) as \(x := (z,t)\), where \(z \in \partial K\), and \(t \in (0, t(z))\) is the distance between \(x\) and \(z\). Write

$$\begin{aligned}&{\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ] = {\lambda }^{2{\beta }} \int _K g(x) \mathbb E \xi (x, \mathcal{P }_{\lambda }\cap K) dx\\&\quad = {\lambda }^{2{\beta }} \int _{z \in { \partial K }} \int _0^{t(z)} g((z,t)) \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K) \cdot \Pi _{i=1}^{d-1} (1 - t C_{z,i}) dt dz. \end{aligned}$$

For each \(z \in { \partial K }\), we apply the transformation \(\mathcal{A }_z\) to \(K\). Recalling from (3.1) that \(\xi \) is stable under \(\mathcal{A }_z\), we have \(\mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K) = \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K_z)\), since \(\mathcal{A }_z(z, t):=(z, t)\) and \(\mathcal{A }_z(\mathcal{P }_{\lambda }\cap K) \stackrel{\mathcal{D }}{=}\mathcal{P }_{\lambda }\cap K_z\). It follows that

$$\begin{aligned} {\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ] \!=\! {\lambda }^{2{\beta }} \!\int _{z \in { \partial K }} \int _0^{t(z)} g((z,t)) \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K_z) \cdot \Pi _{i=1}^{d-1} (1 \!-\! t C_{z,i}) dt dz. \end{aligned}$$

By Lemma 4.1(a), the bound (4.9) with \(p = 2\), and the Cauchy–Schwarz inequality, it follows that uniformly in \(x \in K_z \setminus K_z({\epsilon ^2_{\lambda }})\) we have \(\lim _{{\lambda }\rightarrow \infty } {\lambda }^{2 {\beta }} \mathbb E \xi (x, \mathcal{P }_{\lambda }\cap K_z) = 0\). Since

$$\begin{aligned} \sup _{{\lambda }\ge 1} \sup _{x \in K_z \setminus K_z({\epsilon ^2_{\lambda }}) } {\lambda }^{2 {\beta }} \mathbb E \xi (x, \mathcal{P }_{\lambda }\cap K_z) \le C, \end{aligned}$$

the bounded convergence theorem shows that we can restrict the range of integration of \(t\) to the interval \([0, {\epsilon ^2_{\lambda }}]\) with error \(o(1)\). This gives

$$\begin{aligned} {\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ] = {\lambda }^{2{\beta }} \int _{z \in { \partial K }} \int _0^{{\epsilon ^2_{\lambda }}} g((z,t)) \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K_z ) \cdot \Pi _{i=1}^{d-1} (1 - t C_{z,i}) dt dz + o(1).\nonumber \\ \end{aligned}$$
(5.1)

Changing variables with \(t = r_z (r_z^d {\lambda })^{-2 {\beta }}h\) and using \(h = (r_z^d {\lambda })^{2 {\beta }} (r_z - r)/r_z = (r_z^d {\lambda })^{2 {\beta }}(t/r_z)\) gives \(\xi ((z,t), \mathcal{P }_{\lambda }\cap K_z) = \xi ^{{\lambda }, z}((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}})\). Letting \(h({\lambda }, z):= r_z^{-1 + 2 {\beta }d} {\lambda }^{2 {\beta }} \epsilon _{\lambda }^2\) we obtain

$$\begin{aligned}&{\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ]\\&\quad \!=\! \int _{z \in { \partial K }} r_z^{1 - 2 {\beta }d} \int _0^{h({\lambda }, z) } g((z, o(1))) \mathbb E \xi ^{{\lambda }, z}((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}}) \cdot \Pi _{i=1}^{d-1} (1 - o(1) ) dh dz \!+\! o(1), \end{aligned}$$

where the first \(o(1)\) denotes a quantity tending to zero as \({\lambda }\rightarrow \infty \), uniformly in \(z \in \partial K\) and uniformly in \(h \in [0, h({\lambda }, z)]\).

Note that \((\mathbf{0},h)\) belongs to \(S^{{\lambda }, z} \cap B^{{\lambda }, z}\) and so we may apply Lemma 4.5 to \(\xi ^{{\lambda }, z}((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}})\). Thus, with \({w^{\prime }}\) set to \((\mathbf{0},h)\) in Lemma 4.5, we have

$$\begin{aligned} \sup _{z \in \partial \! K} \sup _{h \in [0, h({\lambda }, z)]} h({\lambda }, z) \left| \mathbb E {\xi ^{{\lambda }, z} }((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}}) - \mathbb E {\xi ^{{\lambda }, z} }((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}_{r_z} }) \right| = o(1), \end{aligned}$$

and so we may replace \(\mathbb E {\xi ^{{\lambda }, z} }((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}})\) by \(\mathbb E {\xi ^{{\lambda }, z} }((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}_{r_z} })\) with error \(o(1)\). We also have \(r_z^{1 - 2 {\beta }d} = {\kappa }(z)^{1/(d + 1)}\). In other words,

$$\begin{aligned}&{\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ] \\&\quad = \int _{z \in { \partial K }} {\kappa }(z)^{1/(d + 1)} \int _0^{h({\lambda }, z) } g((z, o(1)) \mathbb E {\xi ^{{\lambda }, z} }((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}_{r_z} }) \cdot \Pi _{i=1}^{d-1} (1 - o(1) ) dh dz + o(1). \end{aligned}$$

By Lemma 3.2 of [8], the integrand is dominated by an exponentially decaying function of \(h\), uniformly in \(z\) and \({\lambda }\).

The continuity of \(g\), Lemma 4.6, and the dominated convergence theorem give

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-1 + 2{\beta }} \mathbb E [\langle g, \mu _{\lambda }^\xi \rangle ] = \int _{z \in { \partial K }} g(z) {\kappa }(z)^{1/(d + 1)} \int _0^{\infty } \mathbb E \xi ^{(\infty )}((\mathbf{0}, h), \mathcal{P }) dh dz.\nonumber \\ \end{aligned}$$
(5.2)

This gives (2.4), as desired.

5.2 Proof of variance asymptotics (2.5)

Recalling (4.17), for fixed \(g \in \mathcal{C }(K)\) we have

$$\begin{aligned}&{\lambda }^{-1 + 2{\beta }} \mathrm{Var}[\langle g, \mu _{\lambda }^\xi \rangle ] \\&\quad = {\lambda }^{2{\beta }} \int _K g(x)^2 \mathbb E \xi ^2(x, \mathcal{P }_{\lambda }\cap K) dx \\&\quad \quad + \ {\lambda }^{1+ 2{\beta }} \int _{K} \int _{K} g(x) g(y) c(x,y; \mathcal{P }_{\lambda }\cap K) dy dx := I_1({\lambda }) + I_2({\lambda }). \end{aligned}$$

Following the proof of (2.4) until (5.2) shows that

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty }I_1({\lambda }) = \int _{z \in { \partial K }} g(z)^2 {\kappa }(z)^{1/(d + 1)} \int _0^{\infty } \mathbb E (\xi ^{(\infty )}((\mathbf{0}, h), \mathcal{P }))^2 dh dz. \end{aligned}$$
(5.3)

Turning to \(I_2({\lambda })\), write \(x\) in curvilinear coordinates \((z,t)\) with respect to \({ \partial K }\). This gives \(dx = \Pi _{i=1}^{d-1}(1 - t C_{z,i})dt dz\). Apply the map \(\mathcal{A }_z\), write \(\mathcal{A }_z(y) = \bar{y}\) for \(y \in K\), and use stability (3.1) to get

$$\begin{aligned} I_2({\lambda })&= {\lambda }^{1 + 2{\beta }}\int _{z \in { \partial K }} \int _0^{t(z)} \int _{ \bar{y}\in K_z } g((z,t)) g( \bar{y}) c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z) d \bar{y}\cdot \nonumber \\&\times \Pi _{i=1}^{d-1}(1 - t C_{z,i}) dt dz. \end{aligned}$$
(5.4)

Here

$$\begin{aligned}&c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z) = \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K_z \cup \{ \bar{y}\}) \xi ( \bar{y}, \mathcal{P }_{\lambda }\cap K_z \cup \{(z,t)\}) -\\&\quad \mathbb E \xi ((z,t), \mathcal{P }_{\lambda }\cap K_z) \mathbb E \xi ( \bar{y}, \mathcal{P }_{\lambda }\cap K_z ). \end{aligned}$$

The McMullen bound [10] gives

$$\begin{aligned} |c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z)| \le C\mathbb E [N({\lambda })^d] \le C {\lambda }^d, \end{aligned}$$
(5.5)

where here \(N({\lambda })\) denotes the cardinality of \(\mathcal{P }_{\lambda }\cap K_z\).

We make the following three modifications to the triple integral (5.4), each one giving an error of \(o(1)\):

  1. (i)

    Replace the integration domain \(\{ \bar{y}\in K_z \}\) by \(\{ \bar{y}\in K_z({\epsilon ^2_{\lambda }})\}\). Indeed, uniformly in \( \bar{y}\in K_z \setminus K_z({\epsilon ^2_{\lambda }})\) we have

    $$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{1 + 2 {\beta }} c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z) = 0, \end{aligned}$$

    by Lemma 4.1(a), the bound (5.5), and the Cauchy–Schwarz inequality. Since

    $$\begin{aligned} \sup _{{\lambda }\ge 1} \sup _{(z,t), \bar{y}\in K_z \setminus K_z({\epsilon ^2_{\lambda }})} {\lambda }^{1 + 2 {\beta }} c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z) \le C, \end{aligned}$$

    the assertion follow by the bounded convergence theorem.

  2. (ii)

    Replace the integration domain \(\{ \bar{y}\in K_z({\epsilon ^2_{\lambda }})\}\) by \(\{ \bar{y}\in K_z( {\epsilon ^2_{\lambda }}) \cap B_{2D_1 {\epsilon _{\lambda }}}((z,t))\}\) (use Lemma 4.9 and the bounded convergence theorem).

  3. (iii)

    Replace the integration domain \([0,t(z)]\) by \([0,{\epsilon ^2_{\lambda }}]\), as at (5.1). These modifications yield

    $$\begin{aligned} I_2({\lambda })&= {\lambda }^{1 + 2{\beta }} \int _{z \in { \partial K }} \int _0^{{\epsilon ^2_{\lambda }}} \int _{ \bar{y}\in K_z({\epsilon ^2_{\lambda }}) \cap B_{2D_1 {\epsilon _{\lambda }}}((z,t)) } g((z,t)) g( \bar{y}) c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z) d \bar{y}\nonumber \\&\times \Pi _{i=1}^{d-1}(1 - t C_{z,i}) dt dz + o(1). \end{aligned}$$
    (5.6)

Changing variables with \( \bar{y}= (r,u)\) gives \(d \bar{y}= r^{d-1}dr d\sigma _{d-1}(u)\) and it also gives

$$\begin{aligned} \mathcal T ^{{\lambda },z}((r,u))&= \left( \left(r_z^d {\lambda }\right)^{\beta }\exp _{d-1}^{-1}(u), \left(r_z^d {\lambda }\right)^{2 {\beta }}\left(1 - {r \over r_z}\right)\right)\\&= \left(\left(r_z^d {\lambda }\right)^{\beta }v, h^{\prime }\right) = (v^{\prime },h^{\prime }) = {w^{\prime }}. \end{aligned}$$

Thus the covariance \(c((z,t), \bar{y}; \mathcal{P }_{\lambda }\cap K_z)\) transforms to \({ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\), with \({ c^{{\lambda }, z} }\) defined at (4.18). Now change variables with \(t = r_z (r_z^d {\lambda })^{-2{\beta }}h, v^{\prime } = (r_z^d {\lambda })^{{\beta }} v, \) and \(h^{\prime } = (r_z^d {\lambda })^{2{\beta }}(1 - \frac{r}{r_z})\).

The differential \({\lambda }^{1 + 2 {\beta }}\Pi _{i=1}^{d-1}(1 - t C_{z,i}) r^{d-1}dr d \sigma _{d-1}(u) dt dz\) transforms to the differential

$$\begin{aligned}&{\lambda }^{1 + 2 {\beta }} \Pi _{i=1}^{d-1}(1 - r_z (r_z^d {\lambda })^{-2 {\beta }} h C_{z,i}) ((1 - (r_z^d {\lambda })^{-2{\beta }}h^{\prime })r_z)^{d-1} r_z (r_z^d {\lambda })^{-2 {\beta }} dh^{\prime } \\&\quad \times (r_z^d {\lambda })^{-{\beta }(d-1)} dv^{\prime } r_z (r_z^d {\lambda })^{-2{\beta }} dh dz\\&\quad \quad = \Pi _{i=1}^{d-1}(1 - r_z (r_z^d {\lambda })^{-2 {\beta }} h C_{z,i}) (1 - (r_z^d {\lambda })^{-2{\beta }}h^{\prime })^{d-1} r_z^{1 - 2 {\beta }d} dh^{\prime } dv^{\prime } dh dz. \end{aligned}$$

The upper limit of integration \({\epsilon ^2_{\lambda }}\) in (5.6) changes to \(h({\lambda }, z) \) and the domain of integration \(K_z({\epsilon ^2_{\lambda }}) \cap B_{2D_1 {\epsilon _{\lambda }}}((z,t))\) gets mapped to \(S^{{\lambda }, z}\). This gives

$$\begin{aligned} I_2({\lambda }) = \int _{z \in { \partial K }} \int _0^{h({\lambda }, z)} \int _{(v^{\prime },h^{\prime }) \in S^{{\lambda },z} } G_{{\lambda }}(h^{\prime },v^{\prime },h, z) dh^{\prime }dv^{\prime } dh dz + o(1), \end{aligned}$$
(5.7)

where, recalling \(r_z^{1 - 2{\beta }d} = {\kappa }(z)^{1/(d + 1)}\), we get

$$\begin{aligned} G_{{\lambda }}(h^{\prime },v^{\prime }, h, z)&:= {\kappa }(z)^{1/(d + 1)} g((z, o(1))) g(r_z(1- o(1)), (r_z^d {\lambda })^{-{\beta }}v^{\prime }) \\&\times { c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}}) \Pi _{i=1}^{d-1}(1 - o(1)) (1 - o(1))^{d-1}, \end{aligned}$$

where the first \(o(1)\) denotes a term tending to zero uniformly in \((z,h) \in \partial K \times [0,h({\lambda }, z)]\), the second denotes a term tending to zero uniformly in \(r_z, z \in { \partial K },\) and the latter two \(o(1)\) terms denote a quantity tending to zero uniformly in \((v^{\prime },h^{\prime }) \in S^{{\lambda },z}\). We next restrict the integration domain \(S^{{\lambda },z}\) to \(S^{{\lambda },z} \cap B^{{\lambda }, z}\) since by Lemma 4.4 and the moment bounds (4.9) we have

$$\begin{aligned} \int _{z \in { \partial K }} \int _0^{h({\lambda }, z)} \int _{(v^{\prime },h^{\prime }) \in S^{{\lambda },z} \cap (B^{{\lambda }, z})^c } G_{{\lambda }}(h^{\prime },v^{\prime },h, z) dh^{\prime }dv^{\prime } dh dz = o(1). \end{aligned}$$

By Lemma 4.8, uniformly on the range \(\{(v^{\prime },h^{\prime }) \in S^{{\lambda },z} \cap B^{{\lambda }, z} \}\) and uniformly over \(h \in [0, h({\lambda },z)]\), the covariance term \({ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\) differs from the covariance term \({ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}_{r_z} })\) by a term of order \( {\lambda }^{-{\beta }/3}\), modulo logarithmic terms. The integral of this difference over

$$\begin{aligned} (h^{\prime },v^{\prime },h,z) \in S^{{\lambda },z} \times [0, h({\lambda }, z)] \times { \partial K }\end{aligned}$$

is also \(o(1).\) Recalling (5.7), this gives

$$\begin{aligned} I_2({\lambda }) = \int _{z \in { \partial K }} \int _{|h| \le h({\lambda }, z)} \int _{(v^{\prime },h^{\prime }) \in S^{{\lambda },z} \cap B^{{\lambda }, z} } \tilde{G}_{{\lambda }}(h^{\prime },v^{\prime },h, z) dh^{\prime }dv^{\prime } dh dz + o(1), \end{aligned}$$

where

$$\begin{aligned} \tilde{G}_{{\lambda }}(h^{\prime }, v^{\prime }, h, z)&= {\kappa }(z)^{1/(d + 1)} g((z, o(1)) ) g(r_z(1 - o(1)), o(1) )\\&\times { c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}_{r_z} }) \Pi _{i=1}^{d-1} (1 - o(1)) (1 - o(1))^{d-1}. \end{aligned}$$

Recalling the definition of \(\zeta _{\xi ^{(\infty )}}\) at (2.2) we get via Lemma 7.2 of [8] that

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } \tilde{G}_{{\lambda }}(h^{\prime }, v^{\prime }, h, z) = {\kappa }(z)^{1/(d + 1)} g(z)^2 \ \zeta _{\xi ^{(\infty )}}((\mathbf{0},h), (v^{\prime },h^{\prime }); \mathcal{P }). \end{aligned}$$

The first part of Lemma 7.3 of [8] shows that \({ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}_{r_z} })\) is dominated by an integrable function of \(h^{\prime },v^{\prime },h\) and \(z\) on \([0, \infty ) \times \mathbb R ^{d-1} \times [0, \infty ) \times { \partial K }\). Since \(\sup _{z \in { \partial K }} | \ r_z^{d+1}|\) and \(||g||_{\infty }\) are both bounded and since the integration domain \(S^{{\lambda },z} \cap B^{{\lambda }, z} \) increases up to \(\mathbb R ^{d-1} \times [0, \infty )\), the dominated convergence theorem gives

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } I_2({\lambda }) \!=\!\! \int _{z \in { \partial K }} g(z)^2 {\kappa }(z)^{1/(d + 1)} \int _0^{\infty } \int _\mathbb{R ^{d-1}} \int _0^{\infty } \zeta _{\xi ^{(\infty )}}((\mathbf{0},h), (v^{\prime },h^{\prime }); \mathcal{P }) dh^{\prime } dv^{\prime } dh dz.\nonumber \\ \end{aligned}$$
(5.8)

Combining (5.3) and (5.8) gives

$$\begin{aligned}&\lim _{{\lambda }\rightarrow \infty } {\lambda }^{-1 + 2 {\beta }} \mathrm{Var}[\langle g, \mu _{\lambda }^\xi \rangle ] \\ \quad&\quad = \int _{{ \partial K }} g(z)^2 {\kappa }(z)^{1/(d + 1)} \int _0^{\infty } \mathbb E (\xi ^{(\infty )}((\mathbf{0}, h), \mathcal{P }))^2 dh \ + \\&\quad \quad + \ {\kappa }(z)^{1/(d + 1)} \int _0^{\infty } \int _\mathbb{R ^{d-1}} \int _0^{\infty } \zeta _{\xi ^{(\infty )}}((\mathbf{0},h), (v^{\prime },h^{\prime }); \mathcal{P }) dh^{\prime } dv^{\prime } dh dz. \end{aligned}$$

Recalling the definition of \(\sigma ^2(\xi ^{(\infty )})\) at (2.3), this yields

$$\begin{aligned} \lim _{{\lambda }\rightarrow \infty } {\lambda }^{-1 + 2 {\beta }} \mathrm{Var}[\langle g, \mu _{\lambda }^\xi \rangle ] = \sigma ^2(\xi ^{(\infty )}) \int _{{ \partial K }} g(z)^2 {\kappa }(z)^{1/(d + 1)} dz. \end{aligned}$$

This concludes the proof of variance asymptotics and the proof of Theorem 2.1. \(\square \)

Remark If one could show that \({\xi ^{{\lambda }, z} }\) localize in the sense of (4.6), then one could show that the moment bounds of Lemma 4.3 are independent of \({\lambda }\). We expect that our main results would then hold for \(K \in \mathcal{K }^2_+\) by making these three changes: (i) replace the right-hand side of (4.10) with \(o(1)|v^{\prime }|^2\), (ii) in Lemmas 4.5 and 4.8, drop the restrictions \({w^{\prime }}_{\!\!0}, {w^{\prime }}\in S^{{\lambda },z} \cap B^{{\lambda }, z},\) and replace the bounds on the right-hand side of (4.11) and (4.19) with \(o(1)\) bounds, and (iii) show that \( { c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\) decays exponentially in \(|v^{\prime }|\) and \(h^{\prime }\), showing that \(G_{\lambda }(h^{\prime },v^{\prime }, h, z)\) is integrable. We could then directly apply the dominated convergence theorem to \(\mathbb E \xi ^{{\lambda }, z}((\mathbf{0}, h), { \mathcal{P }^{{\lambda }, z}})\) and \({ c^{{\lambda }, z} }((\mathbf{0},h), (v^{\prime },h^{\prime }); { \mathcal{P }^{{\lambda }, z}})\) without needing the error approximations of Lemmas 4.5 and 4.8.

6 Proof of Theorem 1.2

Fix \(K \in \mathcal{K }^3_+\). The image of \(K\) by \(x\longmapsto \mathrm{{vol}}(K)^{-1/d}\cdot x\) is a convex body of unit volume so without loss of generality, we may assume in this section that \(\mathrm{{vol}}(K)=1\). The proof of Theorem 1.2 via Theorem 1.1 is a rewriting of a result previously obtained by Vu (see [20], Proposition 8.1) in the case \(k=0\). For sake of completeness, we include here a proof which does not use large deviation results for \(f_k(K_{\lambda })\).

The method here uses a coupling of the Poisson point process of intensity \(n\) and the binomial point process. Moreover, whether one uses de-Poissonization based on coupling or the approach of [20], the methods in either case establish that much of the limit theory of [8] extends to binomial input. This addresses a technical issue raised at the end of Section 1 of [8].

Let \(X_i\), \(i \ge 1\), be a sequence of i.i.d. uniform random variables in \(K({\epsilon _n^2})\) and put \(\mathcal{X }_n:= \{X_1,\ldots ,X_n\}\). For sake of simplicity, we denote by \(f_k(\mathcal{X }_n \cap K({\epsilon _n^2}))\) the number of \(k\)-dimensional faces of the convex hull of \(\mathcal{X }_n \cap K({\epsilon _n^2})\). In particular, we have

$$\begin{aligned} f_k(\mathcal{X }_n \cap K({\epsilon _n^2})):= \sum _{X_i \in \mathcal{X }_n \cap K({\epsilon _n^2})} \xi _k(X_i, \mathcal{X }_n \cap K({\epsilon _n^2})). \end{aligned}$$

We start with two preliminary lemmas which describe the growth of \(f_k(\mathcal{X }_n\cap K({\epsilon _n^2}))\).

Lemma 6.1

Fix \(K \in \mathcal{K }^3_+\). For all \(k \in \{0,1,\ldots ,d-1 \}\) there is an event \(F(n), P[F(n)^c] = O(n^{-4d})\), and a constant \(C_1 \in (0, \infty )\) such that on \(F(n)\)

$$\begin{aligned} |f_k(\mathcal{X }_n \cap K({\epsilon _n^2})) - f_k(\mathcal{X }_{n + 1} \cap K({\epsilon _n^2}))| \le C_1 (\log n)^{k + 1}. \end{aligned}$$
(6.1)

Proof.

As in the proof of Lemma 4.1 and as on the pages 499–502 of [14], there is an event \(F_1(n)\) with \(P[F_1(n)^c] = O(n^{-4d}),\) such that on \(F_1(n)\) we have for \(X_i \in K({\epsilon _n^2}), \ 1 \le i \le n + 1,\)

$$\begin{aligned} \xi _{k}(X_i, \mathcal{X }_n \cap K({\epsilon _n^2})) = \xi _k(X_i, \mathcal{X }_n \cap K({\epsilon _n^2}) \cap B_{D_1 \epsilon _n}(X_i) ). \end{aligned}$$

It follows that if \(X_i \in B^c_{D_1 \epsilon _n}(X_{n +1}) \cap K({\epsilon _n^2})\), then on \(F_1(n)\) we have

$$\begin{aligned} \xi _k(X_i, \mathcal{X }_n \cap K({\epsilon _n^2})) = \xi _k(X_i, \mathcal{X }_{n + 1} \cap K({\epsilon _n^2})). \end{aligned}$$

Thus on \(F_1(n)\) we have

$$\begin{aligned}&|f_k(\mathcal{X }_n \cap K({\epsilon _n^2})) - f_k(\mathcal{X }_{n + 1} \cap K({\epsilon _n^2}))| \\&\quad \le \xi _k(X_{n + 1}, \mathcal{X }_{n + 1}) + \sum \limits _{X_i \in B_{D_1 \epsilon _n}(X_{n + 1}) \cap K({\epsilon _n^2})} | \xi _k(X_i, \mathcal{X }_n \cap K({\epsilon _n^2})) \\&\qquad - \xi _k(X_i, \mathcal{X }_{n + 1} \cap K({\epsilon _n^2}))|. \end{aligned}$$

The Lebesgue measure of \(B_{D_1 \epsilon _n}(X_{n + 1}) \cap K({\epsilon _n^2})\) is \(O( \epsilon _n^{d-1} {\epsilon _n^2}) = O( \epsilon _n^{d + 1} ) = O( \log n/n)\). There is thus an event \(F_2(n)\), with \(P[F_2^c(n)] = O(n^{-4d})\), such that on \(F_2(n)\) we have

$$\begin{aligned} \text{ card} \{ \mathcal{X }_n \cap B_{D_1 \epsilon _n}(X_{n + 1}) \cap K({\epsilon _n^2}) \} = O (\log n). \end{aligned}$$

The proof of Lemma 4.3 shows that for \(X_i \in B_{D_1 \epsilon _n}(X_{n + 1}) \cap K({\epsilon _n^2})\) there is an event \(F_3(n), P[F_3(n)^c] = O(n^{-4d })\), such that on \(F_3(n)\) we have

$$\begin{aligned} \xi _k(X_i, \mathcal{X }_{n}) = O(( \log n)^k). \end{aligned}$$

The same occurs for \(\xi _k(X_{n+1},\mathcal{X }_{n+1})\). On the event \(F(n): = F_1(n) \cap F_2(n) \cap F_3(n)\) we get (6.1), concluding the proof of Lemma 6.1. \(\square \)

Lemma 6.2

For all \(k \in \{0,1,\ldots ,d-1 \}\) there is a constant \(C_2\) such that for all integers \(l = 1,2,\ldots ,n\) we have

$$\begin{aligned} P[|f_k(\mathcal{X }_n \cap K({\epsilon _n^2})) - f_k(\mathcal{X }_{n + l} \cap K({\epsilon _n^2}))| \ge C_2 l (\log n)^{k + 1}] \le C_2 l n^{-4d}. \end{aligned}$$

Proof.

We have

$$\begin{aligned} |f_k(\mathcal{X }_n \cap K({\epsilon _n^2})) - f_k(\mathcal{X }_{n + l} \cap K({\epsilon _n^2}))|&\le \sum _{i = 0}^{l - 1} |f_k(\mathcal{X }_{n + i} \cap K({\epsilon _n^2})) \nonumber \\&- f_k(\mathcal{X }_{n + i + 1} \cap K({\epsilon _n^2}))|. \end{aligned}$$

By Lemma 6.1, the \(i\)th summand is bounded by \(C_1 (\log (n + i))^{k + 1}\) on a set whose complement probability is \(O(n^{-4d})\). Since \(C_1 (\log (n + i))^{k + 1} \le C (\log 2n)^{k + 1}\), the result follows. \(\square \)

For every \(\lambda >0\), let \(N({\lambda })\) denote a Poisson variable of mean \({\lambda }\) and for every integer \(n\) and \(p\in (0,1)\), let \(\mathrm {Bi}(n,p)\) denote a Binomial variable of parameters \(n\) and \(p\). The next result yields Theorem 1.2.

Theorem 6.1

Fix \(K \in \mathcal{K }^3_+\) and let \(K_n^{\prime }\) be the convex hull of \(n\) independent and uniformly distributed points in \(K\), \(n\ge 1\). For all \(k \in \{0,1,\ldots ,d-1 \}\) we have

$$\begin{aligned} |\mathrm{Var}f_k(K_n^{\prime }) - \mathrm{Var}f_k(K_{N(n)}^{\prime })| = O\left( n^{1 - \frac{3}{d + 1} + o(1)} \right). \end{aligned}$$

Proof.

For all integers \(m\) we put \(H_m := f_k(K_m^{\prime })\). We have

$$\begin{aligned} \mathrm{Var}H_n = \mathrm{Var}H_{N(n)} + \mathrm{Var}(H_n - H_{N(n)}) + 2 \mathrm{Cov}(H_{N(n)}, H_n - H_{N(n)} ). \end{aligned}$$

By (1.2), we have

$$\begin{aligned} \mathrm{Cov}(H_{N(n)}, H_n - H_{N(n)} )&\le \sqrt{ \mathrm{Var}H_{N(n)} } \ ||H_n - H_{N(n)} ||_2 \\ &=O \left( n^{(d-1)/2(d + 1)} \right) ||H_n - H_{N(n)} ||_2. \end{aligned}$$
(6.2)

It is thus enough to show

$$\begin{aligned} || H_n - H_{N(n)} ||_2^2 = O(n^{1 - \frac{4}{d + 1} + o(1)} ) \end{aligned}$$
(6.3)

since then the last two terms in (6.2) are both \(O\left( n^{1 - \frac{3}{d + 1} + o(1)} \right)\).

Given the binomial and Poisson distributions \(\mathcal{L }(\mathrm {Bi}(n, {\epsilon _n^2}))\) and \(\mathcal{L }(N(n {\epsilon _n^2}))\), there exist coupled random variables \(\mathrm {Bi}(n, {\epsilon _n^2})\) and \(N(n {\epsilon _n^2})\) such that

$$\begin{aligned} P[ \mathrm {Bi}(n, {\epsilon _n^2}) \ne N(n {\epsilon _n^2})] \le {\epsilon _n^2}; \end{aligned}$$
(6.4)

see e.g. (1.4) and (1.23) of [5].

Enumerate the points \(\mathcal{P }_n \cap K({\epsilon _n^2})\) by \(X_1, X_2,\ldots ,X_{N({\epsilon _n^2})}\). Given \(\mathrm {Bi}(n, {\epsilon _n^2})\), consider the coupled point set \(\mathcal{Y }_n\) obtained by discarding or adding i.i.d. points \(X_i\) in \(K({\epsilon _n^2})\):

$$\begin{aligned} \mathcal{Y }_n : = \left\{ \begin{array}{l@{\quad }l} X_1,\ldots ,X_{N({\epsilon _n^2})- (N({\epsilon _n^2}) - \mathrm {Bi}(n, {\epsilon _n^2}) )^+} ,&\mathrm{if} \ N({\epsilon _n^2}) \ge \mathrm {Bi}(n, {\epsilon _n^2})\\ X_1,\ldots ,X_{N({\epsilon _n^2}) + (\mathrm {Bi}(n, {\epsilon _n^2}) - N({\epsilon _n^2}))^+} ,&\mathrm{if} \ N({\epsilon _n^2}) < \mathrm {Bi}(n, {\epsilon _n^2}). \end{array} \right. \end{aligned}$$

Then \(\mathcal{Y }_n \stackrel{\mathcal{D }}{=}\mathcal{X }_n \cap K({\epsilon _n^2}) = X_1, X_2,\ldots ,X_\mathrm{{Bi}(n, {\epsilon _n^2})}.\) We use this coupling of the point sets \(\mathcal{P }_n \cap K({\epsilon _n^2})\) and \(\mathcal{X }_n \cap K({\epsilon _n^2})\) in all that follows.

Denoting the convex hull of \(m\) i.i.d. points \(X_1,\ldots ,X_m\) on \(K(s)\) by \(K(s)^{\prime }_m\), we have

$$\begin{aligned} || H_n - H_{N(n)} ||_2^2&= \int (f_k(K_n^{\prime }) - f_k(K_{N(n)}^{\prime }))^2 dP \\&\!=\! \int \left[ f_k(K({\epsilon _n^2})_{ \mathrm{{Bi}}(n, \mathrm{{vol}}(K({\epsilon _n^2}))}^{\prime }) \!-\! f_k (K({\epsilon _n^2})_{N(n \mathrm{{vol}}(K({\epsilon _n^2})))}^{\prime }) \right]^2 dP \!+\! o(1), \end{aligned}$$

where the last equality follows from the \(O(n^{-4d})\) probability bounds of Lemma 4.1(a), the bounds \(f_k(K({\epsilon _n^2})_{ j}^{\prime }) \le C_3j^{d/2}\), and the Cauchy–Schwarz inequality. Let \(E_n\!:=\! \{Bi(\text{ n}, \text{ vol}(K({\epsilon _n^2}))) \!\ne \! N(\text{ n} \text{ vol}(K({\epsilon _n^2}))) \}\) and recall from (6.4) that \(P[E_n] \le {\epsilon _n^2}\). On \(E_n^c\) the integrand vanishes. Thus

$$\begin{aligned}&|| H_n - H_{N(n)} ||_2^2 \\&\quad = \int \left[ f_k(K({\epsilon _n^2})_{ \mathrm {Bi}(n,\mathrm {vol}(K({\epsilon _n^2})))} ) - f_k (K({\epsilon _n^2})_{N(n \mathrm {vol}(K({\epsilon _n^2})))} ) \right]^2 \mathbf{1}(E_n) dP + o(1). \end{aligned}$$

By the Bernstein inequality there is a constant \(C_4\) such that for all \(p \in (0, 1/2)\) we have

$$\begin{aligned} |\mathrm{{Bi}}(n, p) - np| \le C_4 (\log (np)) \sqrt{np} \end{aligned}$$

with probability at least \(1 - O(n^{-4d})\). By Proposition A.2.3 of [5], and taking \(C_4\) larger if necessary, we also have

$$\begin{aligned} |\mathrm{{N}}(np) - np| \le C_4 (\log (np)) \sqrt{np} \end{aligned}$$

with probability at least \(1 - O(n^{-4d})\). A modification of Lemma 6.2 shows that there is an event \(G_n(1)\) with probability at least \(1 - O((\log n)^{1 +1/(d+1)}n^{1/2-1/(d+1)-4d})\) such that on \(G_n(1)\) we have

$$\begin{aligned} |f_k(K({\epsilon _n^2})_{ \mathrm {Bi}(n,\mathrm {vol}(K({\epsilon _n^2})))} ) - f_k (K({\epsilon _n^2})_{n \lfloor \mathrm {vol}(K({\epsilon _n^2}))\rfloor } )|^2 = O( (\log n)^{2k + 4} n {\epsilon _n^2}). \end{aligned}$$

Similarly, there is an event \(G_n(2)\) with probability at least \(1 - O((\log n)^{1 +1/(d+1)}\) \(n^{1/2-1/(d+1)-4d})\) such that on \(G_n(2)\) we have

$$\begin{aligned} |f_k(K({\epsilon _n^2})_{ \mathrm {N}(n \mathrm {vol}(K({\epsilon _n^2})))} ) - f_k (K({\epsilon _n^2})_{n \lfloor \mathrm {vol}(K({\epsilon _n^2}))\rfloor } )|^2 = O( (\log n)^{2k + 4} n {\epsilon _n^2}). \end{aligned}$$

On the event \(G_n:= G_n(1) \cup G_n(2)\) we have

$$\begin{aligned} |f_k(K({\epsilon _n^2})_{ \mathrm {Bi}(n, {\epsilon _n^2})} ) - f_k (K({\epsilon _n^2})_{ \mathrm {N}(n {\epsilon _n^2})} )|^2 = O( (\log n)^{2k + 4} n {\epsilon _n^2}). \end{aligned}$$
(6.5)

By McMullen’s bound [10]

$$\begin{aligned} |f_k(K({\epsilon _n^2})_{ \mathrm {Bi}(n, {\epsilon _n^2})} ) - f_k (K({\epsilon _n^2})_\mathrm{{N}(n {\epsilon _n^2})} )|^2 \le C_3 ( \mathrm {Bi}(n, {\epsilon _n^2})^{d} + \mathrm {N}(n {\epsilon _n^2})^{d}) \end{aligned}$$

always holds. It follows by the Cauchy–Schwarz inequality that

$$\begin{aligned} \int \left[ f_k(K({\epsilon _n^2})_{ \mathrm {Bi}(n, {\epsilon _n^2})} ) - f_k (K({\epsilon _n^2})_{N(n {\epsilon _n^2})} ) \right]^2 \mathbf{1}(E_n) \mathbf{1}(G^c_n)dP = o(1), \end{aligned}$$

whence in view of (6.5)

$$\begin{aligned} || H_n - H_{N(n)} ||_2^2 = O \left( (\log n)^{2k + 4} n {\epsilon _n^2} \int \mathbf{1}(E_n) \mathbf{1}(G_n)dP \right) + o(1). \end{aligned}$$

It follows that

$$\begin{aligned} || H_n - H_{N(n)} ||_2^2 = O((\log n)^{2k + 4} n {\epsilon _n^2} P[E_n]) + o(1) = O((\log n)^{2k + 4} n \epsilon _n^4 ) + o(1). \end{aligned}$$

This shows (6.3) and concludes the proof of Theorem 6.1. \(\square \)