Abstract
A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge–Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.
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1 Introduction and statement of results
The classical Steiner formula states that the volume of the outer parallel set of a convex body (that is, a non-empty, compact, convex set) in \({\mathbb {R}}^n\) at distance \(r>0\) can be expressed as a polynomial in r of degree at most n. Using that the outer parallel set of a convex body \(K\subset {\mathbb {R}}^n\) at distance \(r>0\) is just the Minkowski (or vector sum) of K and \(rB^n\), the ball of radius r, we get
for every \(r>0\), where \(V_n\) is n-dimensional volume or Lebesgue measure and \(\kappa _j\) is the j-dimensional volume of the unit ball in \({\mathbb {R}}^j\) (with the convention that \(\kappa _0:=1\)). The coefficients \(V_j(K)\) are known as the intrinsic volumes of K. Up to normalization and numbering, they coincide with the classical quermassintegrals. In particular, \(V_{n-1}(K)\) is proportional to the surface area of K and \(V_0(K)\) is the Euler characteristic of K (that is, \(V_0(K):= 1\)) for every convex body K in \({\mathbb {R}}^n\) (cf. [43]).
A complete characterization of intrinsic volumes is due to Hadwiger, who in his celebrated theorem classified all continuous, translation and rotation invariant valuations on the space, \({\mathcal {K}}^n\), of convex bodies in \({\mathbb {R}}^n\). Here, we say that \({{\,\mathrm{{\text {Z}}}\,}}:{\mathcal {K}}^n\rightarrow {\mathbb {R}}\) is a valuation if
for every \(K,L\in {\mathcal {K}}^n\) such that also \(K\cup L\in {\mathcal {K}}^n\). It is translation invariant if \({{\,\mathrm{{\text {Z}}}\,}}(\tau K)={{\,\mathrm{{\text {Z}}}\,}}(K)\) for every \(K\in {\mathcal {K}}^n\) and translation \(\tau \) on \({\mathbb {R}}^n\) and rotation invariant if \({{\,\mathrm{{\text {Z}}}\,}}(\vartheta K)={{\,\mathrm{{\text {Z}}}\,}}(K)\) for every \(K\in {\mathcal {K}}^n\) and \(\vartheta \in {\text {SO}}(n)\). The topology of \({\mathcal {K}}^n\) is induced by the Hausdorff metric.
Theorem 1.1
(Hadwiger [25]) A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathcal {K}}^n\rightarrow {\mathbb {R}}\) is a continuous, translation and rotation invariant valuation if and only if there exist constants \(\zeta _0, \dots , \zeta _n\in {\mathbb {R}}\) such that
for every \(K\in {\mathcal {K}}^n\).
In addition to its many applications in convex and integral geometry (see [25, 27]), the Hadwiger theorem can be used to give a simple proof of (1.1).
We remark that the classification of valuations on convex bodies is a classical subject, which is described in [43, Chapter 6]. Also see [10, 26] for some newly defined valuations and [2, 3, 5, 7, 8, 23, 24, 30, 31, 34, 35] for some recent classification results.
Recently, the authors [16] introduced functional intrinsic volumes on convex functions. Let
denote the space of all proper, super-coercive, lower semicontinuous, convex functions on \({\mathbb {R}}^n\), where \(\vert \cdot \vert \) denotes the Euclidean norm. For \(\zeta \in C_b((0,\infty ))\), the set of continuous functions with bounded support on \((0,\infty )\), and \(0\le j \le n\), consider the functional
on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C_+^2({\mathbb {R}}^n)\), where \(C_+^2({\mathbb {R}}^n)\) denotes the set of \(u\in C^2({\mathbb {R}}^n)\) with positive definite Hessian matrix \({{\text {D}}}^2u\) and \([A]_k\) is the kth elementary symmetric function of the eigenvalues of the symmetric \(n\times n\) matrix A (with the convention that \([A]_0:=1\)).
Under suitable conditions on the function \(\zeta \), the functional (1.2) continuously extends to the whole space \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). Here, continuity is understood with respect to epi-convergence (see Sect. 3.2). In case \(\zeta \) can be identified with an element of \(C_c([0,\infty ))\), the set of continuous functions with compact support on \([0,\infty )\), it was shown in [14] that (1.2) continuously extends to \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) by using Hessian measures (see Sect. 5.2 for the definition).
More recently, the authors proved that (1.2) continuously extends for the following classes of singular densities \(\zeta \). For \(0\le j \le n-1\), let
In addition, let \(D_{n}^{n}\) be the set of all functions \(\zeta \in C_b((0,\infty ))\) such that \(\lim _{s\rightarrow 0^+} \zeta (s)\) exists and is finite. For \(\zeta \in D_{n}^{n}\), we set \(\zeta (0):=\lim _{s\rightarrow 0^+} \zeta (s)\) and identify \(\zeta \) with the corresponding element of \(C_c([0,\infty ))\).
Theorem 1.2
([16], Theorem 1.2) For \(0\le j \le n\) and \(\zeta \in D_{j}^{n}\), there exists a unique, continuous, epi-translation and rotation invariant valuation \({\text {V}}_{j,\zeta }:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) such that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C_+^2({\mathbb {R}}^n)\).
Here, we say that \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is a valuation if
for every \(u,v\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) such that also their pointwise maximum \(u\vee v\) and minimum \(u\wedge v\) belong to \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). A valuation \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is said to be epi-translation invariant if \({{\,\mathrm{{\text {Z}}}\,}}(u\circ \tau ^{-1}+\gamma )={{\,\mathrm{{\text {Z}}}\,}}(u)\) for every translation \(\tau \) on \({\mathbb {R}}^n\), every \(\gamma \in {\mathbb {R}}\) and every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and it is rotation invariant if \({{\,\mathrm{{\text {Z}}}\,}}(u\circ \vartheta ^{-1})={{\,\mathrm{{\text {Z}}}\,}}(u)\) for every \(\vartheta \in {\text {SO}}(n)\) and \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). We remark that these properties are natural extensions of the corresponding properties of the classical intrinsic volumes.
A closed representation of the extensions of (1.3) to \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) was obtained for the cases \(j=0\) and \(j=n\). For \(\zeta \in D_{0}^{n}\), the functional \({\text {V}}_{0,\zeta }\) is a constant, independent of \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), and for \(\zeta \in D_{n}^{n}\), we have
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), where \({\text {dom}}u:=\{x\in {\mathbb {R}}^n:u(x)<\infty \}\) is the domain of u (see [14, Theorem 2]). However, apart from these extremal cases, the functionals \({\text {V}}_{j,\zeta }\) were so far only described as continuous extensions of (1.3) and by Cauchy–Kubota formulas, which were recently established in [17, Theorem 1.6].
In [16], the following functional Hadwiger theorem was established.
Theorem 1.3
([16], Theorem 1.3) A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is a continuous, epi-translation and rotation invariant valuation if and only if there exist functions \(\zeta _0\in D_{0}^{n}\), ..., \(\zeta _n\in D_{n}^{n}\) such that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\).
Using the notion of epi-homogeneity of degree j (see Sect. 3.2), we see that Theorems 1.1 and 1.3 imply that for \(0\le j\le n\), the functionals \({\text {V}}_{j,\zeta }\) for \(\zeta \in D_{j}^{n}\) correspond to multiples of the classical intrinsic volumes \(V_j\). Hence, we call \({\text {V}}_{j,\zeta }\) for \(0\le j\le n\) and \(\zeta \in D_{j}^{n}\) a jth functional intrinsic volume. Moreover, the family \(\{{\text {V}}_{j,\zeta }:\zeta \in D_{j}^{n}\}\) describes all continuous, epi-translation and rotation invariant valuations on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) that are epi-homogeneous of degree j and is, in this sense, canonical.
We remark that the classification of valuations on function spaces has only been started to be studied recently. The first classification results for valuations on classical function spaces were obtained for \(L_p\) and Sobolev spaces, and for Lipschitz and continuous functions (see [18, 19, 32, 33, 48, 49]). Results on valuations on convex functions can be found in [4, 12,13,14, 28, 29, 37, 38].
In this article we present a new, complete family of Steiner formulas for functional intrinsic volumes and its applications. For \(\zeta \in D_{n}^{n}\) (or equivalently, \(\zeta \in C_c([0,\infty ))\)), the functional Steiner formula is the following result.
Theorem 1.4
If \(\zeta \in D_{n}^{n}\), then
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(r>0\), where \(\zeta _j\in D_{j}^{n}\) is given by
for \(s>0\) and \(0\le j \le n\).
Here, \(u\mathbin {\Box }w\) is the infimal convolution of \(u,w\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and 
is obtained by epi-multiplication of w with \(r>0\) while \(\mathbf{I}_{B^n}\) is the convex indicator function of the Euclidean unit ball \(B^n\) (see Sect. 3.2 for the precise definitions). Note that
where \({\text {epi}}w:=\{(x,t)\in {\mathbb {R}}^n\times {\mathbb {R}}: t\ge w(x)\}\) is the epi-graph of \(w:{\mathbb {R}}^n\rightarrow (-\infty ,+\infty ]\) and the addition on the right side of (1.7) is Minkowski addition in \({\mathbb {R}}^n\times {\mathbb {R}}\).
We give two proofs of Theorem 1.4. In Sect. 7, we give a direct proof (not using the functional Hadwiger theorem, Theorem 1.3) and in Sect. 8, we prove Theorem 1.4 using Theorem 1.3. This corresponds to the fact that the classical Steiner formula can be proved both directly and as a consequence of the Hadwiger theorem.
Equation (1.5) corresponds to the classical Steiner formula (1.1). Indeed, we will see that (1.1) can be easily retrieved from (1.5). Furthermore, by properties of the transform (1.6), every functional intrinsic volume \({\text {V}}_{j,\zeta _j}\) for \(1\le j\le n\) and \(\zeta _j\in D_{j}^{n}\) will appear exactly once on the right side of (1.5) as \(\zeta \) ranges in \(D_{n}^{n}\). In this sense, Theorem 1.4 provides a complete description of functional intrinsic volumes on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). We remark that Steiner formulas for convex functions are also obtained if we replace \(\mathbf{I}_{B^n}\) in (1.5) by other radially symmetric, super-coercive, convex functions. However, in general such formulas do not give rise to all functional intrinsic volumes. For more details, see Sect. 8.4.
As an immediate consequence of Theorem 1.4, eq. (1.4) and properties of the transform (1.6) (see Lemma 3.2), we obtain the following new representation of the functionals \({\text {V}}_{j,\zeta }\).
Corollary 1.5
If \(\,0\le j < n\) and \(\zeta \in D_{j}^{n}\), then
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), where \(\alpha \in C_c([0,\infty ))\) is given by
for \(s>0\).
Using a new family of measures, we establish new closed representations of the functional intrinsic volumes on the whole space \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) that do not require singular densities. For \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), let \(\mathrm{MA}^{\!*}(u;\cdot )\) be the push-forward through \(\nabla u\) of n-dimensional Lebesgue measure restricted to the domain of u. Equivalently, \(\mathrm{MA}^{\!*}(u;\cdot )\) is the Monge–Ampère measure of the convex conjugate of u (see Sect. 5 for details) and we call it the conjugate Monge–Ampère measure of u. For functions \(u_1,\dots ,u_n\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), we write \(\mathrm{MA}^{\!*}(u_1,\dots ,u_n;\cdot )\) for the polarization of \(\mathrm{MA}^{\!*}(u;\cdot )\) with respect to infimal convolution (see Sect. 5) and call \(\mathrm{MA}^{\!*}(u_1,\dots ,u_n;\cdot )\) the conjugate mixed Monge–Ampère measure of \(u_1,\dots ,u_n\). For \(0\le j \le n\) and \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), we set
where the function u is repeated j times and the convex indicator function \(\mathbf{I}_{B^n}\) is repeated \((n-j)\) times. We establish the following result.
Theorem 1.6
If \(\,0\le j \le n\) and \(\zeta \in D_{j}^{n}\), then
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), where \(\alpha \in C_c([0,\infty ))\) is given by
for \(s>0\). Moreover, for \(1\le j \le n\),
for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C^2_+({\mathbb {R}}^n)\).
Here, for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C_+^2({\mathbb {R}}^n)\) and \(0\le i\le n-1\), we write \(\tau _i(u,x)\) for the ith elementary symmetric function of the principal curvatures of the sublevel set \(\{y\in {\mathbb {R}}^n:u(y)\le t\}\) at \(x\in {\mathbb {R}}^n\) with \(t=u(x)\) (and we use the convention \(\tau _0(u,x):=1\)). Note that \(\tau _i(u,x)\) is well-defined for such u if \(u(x)> \min _{y\in {\mathbb {R}}^n} u(y)\). Since such u attains its minimum at only one point, the integral in (1.9) is also well-defined. We remark that a direct proof of (1.9) was given in [17, Lemma 3.9]. Here it is a consequence of properties of the measures \(\mathrm{MA}^{*}_{j}(u;\cdot )\) (see Theorem 5.5).
Conjugate mixed Monge–Ampère measures generalize Hessian measures on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) (see Sect. 5.2) and the precise connection of integrals involving the measure \(\mathrm{MA}^{*}_{j}(u;\cdot )\) and Hessian measures for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) is established in Sect. 6. It is the basis of a new proof of Theorem 1.2 presented in Sect. 7, where we also prove Theorem 1.6.
Combining Theorems 1.3 and 1.6, we obtain the following new version of the Hadwiger theorem for convex functions.
Theorem 1.7
A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is a continuous, epi-translation and rotation invariant valuation if and only if there exist functions \(\alpha _0,\dots ,\alpha _n\in C_c([0,\infty ))\) such that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\).
By properties of the integral transform from Theorem 1.6 which maps \(\zeta \) to \(\alpha \), this version is equivalent to Theorem 1.3.
Using the Legendre–Fenchel transform or convex conjugate, we can translate the new results on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) to results on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}:=\{v:{\mathbb {R}}^n\rightarrow {\mathbb {R}}:v \text { is convex}\}\), the space of finite-valued convex functions on \({\mathbb {R}}^n\). In fact, most results will be proved on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and then transferred to \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) using convex conjugation. Results on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) are presented in Sect. 2. The next section is devoted to notation and preliminaries. In Sect. 4, results on Monge–Ampère measures and mixed Monge–Ampère measures on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) are collected and the new measures \(\mathrm{MA}_{j}(v;\cdot )\) for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(0\le j\le n\) are discussed. In Sect. 5, the corresponding results are presented on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). Results connecting the measure \(\mathrm{MA}_{j}(v;\cdot )\) to the jth Hessian measure of \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) are established in Sect. 6. In the following section, the proofs of the main results are presented. In the final section, an alternate proof of the functional Steiner formula, results on the explicit representation of functional intrinsic volumes and on the retrieval of classical intrinsic volumes are presented. Moreover, general functional Steiner formulas are discussed.
2 Results for valuations on finite-valued convex functions
A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is dually epi-translation invariant if and only if \({{\,\mathrm{{\text {Z}}}\,}}(v+\ell +\gamma )={{\,\mathrm{{\text {Z}}}\,}}(v)\) for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), every linear functional \(\ell :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and every \(\gamma \in {\mathbb {R}}\), or equivalently, if the map \(u\mapsto {{\,\mathrm{{\text {Z}}}\,}}(u^*)\), defined on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), is epi-translation invariant. It was shown in [15] that \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is a continuous valuation if and only if \(u\mapsto {{\,\mathrm{{\text {Z}}}\,}}(u^*)\) is a continuous valuation on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) (see Proposition 3.1).
The following result is equivalent to Theorem 1.2 by duality.
Theorem 2.1
([16], Theorem 1.4) For \(0\le j \le n\) and \(\zeta \in D_{j}^{n}\), there exists a unique, continuous, dually epi-translation and rotation invariant valuation \({\text {V}}_{j,\zeta }^{*}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) such that
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2_+({\mathbb {R}}^n)\).
Here, for \(0\le j\le n\) and \(\zeta \in D_{j}^{n}\), the valuation \({\text {V}}_{j,\zeta }^{*}\) is dual to \({\text {V}}_{j,\zeta }\) in the sense that \({\text {V}}_{j,\zeta }^{*}(v)={\text {V}}_{j,\zeta }(v^*)\) for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). We remark that the new proof of Theorem 2.1 that we present in Sect. 7 actually shows that the representation (2.1) holds on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\).
The Hadwiger Theorem on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) is the following result, which is equivalent to Theorem 1.3 by duality.
Theorem 2.2
([16], Theorem 1.5) A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is a continuous, dually epi-translation and rotation invariant valuation if and only if there exist functions \(\zeta _0\in D_{0}^{n}\), ..., \(\zeta _n\in D_{n}^{n}\) such that
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
We obtain the following dual version of the functional Steiner formulas from Theorem 1.4. We use the support function of the unit ball, \(h_{B^n}(x)=|x|\) for \(x\in {\mathbb {R}}^n\), and the fact that 
for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(r>0\).
Theorem 2.3
If \(\zeta \in D_{n}^{n}\), then
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(r>0\), where \(\zeta _j\in D_{j}^{n}\) is given by
for \(s>0\) and \(\,0\le j \le n\).
An immediate consequence is the following result.
Corollary 2.4
Let \(0\le j < n\). If \(\zeta \in D_{j}^{n}\), then
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), where \(\alpha \in C_c([0,\infty ))\) is given by
for \(s>0\).
Let \(\mathrm{MA}(v;\cdot )\) be the Monge–Ampère measure of \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and write \(\mathrm{MA}(v_1,\dots ,v_n;\cdot )\) for its polarization, the mixed Monge–Ampère measure of \(v_1,\dots ,v_n\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). For \(0\le j \le n\) and \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), we set
(see Sect. 4 for results on Monge–Ampère measures, mixed Monge–Ampère measures and this new family of measures).
The following result corresponds to Theorem 1.6.
Theorem 2.5
If \(\,0\le j \le n\) and \(\zeta \in D_{j}^{n}\), then
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), where \(\alpha \in C_c([0,\infty ))\) is given by
for \(s>0\). Moreover, for \(1\le j \le n\),
for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\).
Here, \(\det (A_1,\dots ,A_n)\) denotes the mixed discriminant of the symmetric \(n\times n\) matrices \(A_1,\dots , A_n\). Note that \({{\text {D}}}^2h_{B^n}(x)\) exists for every \(x\ne 0\) and that (2.3) is well-defined as a Lebesgue integral. Combining Theorems 2.2 and 2.5, we obtain the following new version of the Hadwiger theorem for finite-valued convex functions.
Theorem 2.6
A functional \(\,{{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is a continuous, dually epi-translation and rotation invariant valuation if and only if there exist functions \(\alpha _0,\ldots ,\alpha _n\in C_c([0,\infty ))\) such that
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
By properties of the integral transform from Theorem 2.5 which maps \(\zeta \) to \(\alpha \), this version is equivalent to Theorem 2.2.
3 Preliminaries
We work in n-dimensional Euclidean space \({\mathbb {R}}^n\), with \(n\ge 1\), endowed with the Euclidean norm \(\vert \cdot \vert \) and the usual scalar product \(\langle \cdot ,\cdot \rangle \). We also use coordinates, \(x=(x_1,\dots ,x_n)\), for \(x\in {\mathbb {R}}^n\). Let \(B^n:=\{x\in {\mathbb {R}}^n:\vert x\vert \le 1\}\) be the Euclidean unit ball and \({\mathbb {S}^{n-1}}\) the unit sphere in \({\mathbb {R}}^n\). A basic reference on convex bodies is the book by Schneider [43].
3.1 Mixed discriminants
We will need some basic definitions and properties which can be found in Sect. 5.5 of the book by Schneider [43]. Given symmetric \(n\times n\) matrices \(A_k = (a_{ij}^k)\) for \(1\le k \le n\), their mixed discriminant is defined as
where we sum over all permutations \(\sigma \) of \(\{1,\ldots ,n\}\). As a consequence of this definition, the mixed discriminant \(\det \) is multilinear and symmetric in its entries. Alternatively, the mixed discriminant is uniquely determined as the symmetric functional that satisfies
for all \(\lambda _1,\ldots ,\lambda _m\in {\mathbb {R}}\), symmetric \(n\times n\) matrices \(A_1,\ldots ,A_m\) and \(m\ge 1\). By the polarization formula, the mixed determinant can also be written as
for symmetric \(n\times n\) matrices \(A_1,\ldots ,A_n\) (see, for example, [6, Theorem 4]). In addition, there exist maps \(D_{ij}:({\mathbb {R}}^{n\times n})^{n-1}\rightarrow {\mathbb {R}}\) for \(1\le i,j\le n\) such that
for all symmetric \(n\times n\) matrices \(A_1,\ldots ,A_n\). We remark that it follows from (3.1) that
for every symmetric \(n\times n\) matrix A, where \(I_n\) is the \(n\times n\) identity matrix. If the symmetric matrix A is, in addition, invertible, then
for \(0\le j\le n\).
3.2 Convex functions
We collect some basic results and properties of convex functions. Standard references are the books by Rockafellar [41] and Rockafellar & Wets [42].
Let \({\mathrm{Conv}({\mathbb {R}}^n)}\) be the set of proper, lower semicontinuous, convex functions \(u:{\mathbb {R}}^n\rightarrow (-\infty ,\infty ]\), where u is proper if \(u\not \equiv +\infty \). For \(t\in {\mathbb {R}}\), we write
for the sublevel sets of u. If \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), then u attains its minimum and we set
This is a convex body which, if in addition \(u\in C^2_+({\mathbb {R}}^n)\), consists of a single point.
The standard topology on \({\mathrm{Conv}({\mathbb {R}}^n)}\) and its subsets is induced by epi-convergence. A sequence of functions \(u_k\in {\mathrm{Conv}({\mathbb {R}}^n)}\) is epi-convergent to \(u\in {\mathrm{Conv}({\mathbb {R}}^n)}\) if for every \(x\in {\mathbb {R}}^n\):
-
(i)
\(u(x)\le \liminf _{k\rightarrow \infty } u_k(x_k)\) for every sequence \(x_k\in {\mathbb {R}}^n\) that converges to x;
-
(ii)
\(u(x)=\lim _{k\rightarrow \infty } u_k(x_k)\) for at least one sequence \(x_k\in {\mathbb {R}}^n\) that converges to x.
Note that the limit of an epi-convergent sequence of functions from \({\mathrm{Conv}({\mathbb {R}}^n)}\) is always lower semicontinuous.
For \(u\in {\mathrm{Conv}({\mathbb {R}}^n)}\), we denote by \(u^*\in {\mathrm{Conv}({\mathbb {R}}^n)}\) its Legendre–Fenchel transform or convex conjugate, which is defined by
for \(y\in {\mathbb {R}}^n\). Since u is lower semicontinuous, we have \(u^{**}=u\). For a convex body \(K\in {\mathcal {K}}^n\), we denote by \(\mathbf{I}_K\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) its convex indicator function, which is defined as
We have
where \(h_K:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is the support function of K, defined as
For \(u_1,u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), we denote by \(u_1\mathbin {\Box }u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) their infimal convolution or epi-sum which is defined as
for \(x\in {\mathbb {R}}^n\). The epi-multiplication of \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) by \(\lambda >0\) is defined in the following way. We set
for \(x\in {\mathbb {R}}^n\) and note that 
. This corresponds to rescaling the epi-graph of u by the factor \(\lambda \), that is, 
.
Proposition 3.1
The following properties hold.
-
(a)
The function \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) if and only if \(u^*\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
-
(b)
The function \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C^2_+({\mathbb {R}}^n)\) if and only if \(u^*\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C^2_+({\mathbb {R}}^n)\).
-
(c)
If \(u_1, u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) are such that \(u_1\vee u_2\) and \(u_1\wedge u_2\) are in \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), then \(u_1^*\vee u_2^*\) and \( u_1^*\wedge u_2^*\) are in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and
$$\begin{aligned} (u_1\vee u_2)^*=u_1^*\wedge u_2^*,\quad \quad (u_1\wedge u_2)^*=u_1^*\vee u_2^*. \end{aligned}$$ -
(d)
For \(u_1, u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(\lambda _1,\lambda _2>0\),
-
(e)
The sequence \(u_k\) in \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) epi-converges to \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), if and only if the sequence \(u^*_k\) in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) epi-converges to \(u^*\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
We say that a functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is epi-homogeneous of degree j if
for every \(\lambda >0\) and \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). A functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is homogeneous of degree j if
for every \(\lambda >0\) and \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). It is a consequence of Proposition 3.1 that a map\({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is a continuous valuation that is epi-homogeneous of degree j if and only if \(v\mapsto {{\,\mathrm{{\text {Z}}}\,}}(v^*)\) is a continuous valuation on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) that is homogeneous of degree j. We say that \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\) is epi-additive if
for every \(u_1, u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). The dual notion is additivity on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), where a functional \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is additive if
for every \(v_1, v_2\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
3.3 The integral transform \({\mathcal {R}}\)
In [17], the integral transform \({\mathcal {R}}\) and its inverse \({\mathcal {R}}^{-1}\) were introduced. For \(\zeta \in C_b((0,\infty ))\) and \(s>0\), let
It is easy to see that also \({\mathcal {R}}\zeta \in C_b((0,\infty ))\).
For \(l\ge 1\), we write
and set \({\mathcal {R}}^0 \zeta := \zeta \). We set \({\mathcal {R}}^{-l}=({\mathcal {R}}^{-1})^l\) for \(l\ge 1\). We require the following result.
Lemma 3.2
([17], Lemmas 3.5 and 3.7) For \(0\le k \le n\) and \(0\le l \le n-k\), the map \({\mathcal {R}}^l:D_{k}^{n}\rightarrow D_{k}^{n-l}\) is a bijection. Furthermore,
for every \(\zeta \in D_{k}^{n}\) and \(s>0\), while
for every \(\rho \in D_{k}^{n-l}\) and \(s>0\).
For \(t\ge 0\), let \(u_t\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) be given by
for \(x\in {\mathbb {R}}^n\). The next result shows that the transform \({\mathcal {R}}\) naturally occurs when studying functional intrinsic volumes.
Lemma 3.3
([16], Lemmas 2.15 and 3.24) If \(\,1\le j \le n\) and \(\zeta \in D_{j}^{n}\), then
for \(t\ge 0\).
We also require the dual form of the previous result. For \(t\ge 0\), we set \(v_t:=u_t^*\). Note that
for \(x\in {\mathbb {R}}^n\) and \(t\ge 0\) and that \(v_t\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) for \(t\ge 0\).
Lemma 3.4
([16]) If \(\,1\le j \le n\) and \(\zeta \in D_{j}^{n}\), then
for \(t\ge 0\).
4 Monge–Ampère and mixed Monge–Ampère measures
For \(w\in {\mathrm{Conv}({\mathbb {R}}^n)}\), the subdifferential of w at \(x\in {\mathbb {R}}^n\) is defined by
Each element of \(\partial w(x)\) is called a subgradient of w at x. If w is differentiable at x, then \(\partial w(x)=\{\nabla w(x)\}\). Given a subset A of \({\mathbb {R}}^n\), we define the image of A through the subdifferential of w as
We write \(\vert {\,\cdot \,}\vert \) for n-dimensional Lebesgue measure in \({\mathbb {R}}^n\) and remark that \(\vert {\partial w(C)}\vert \) can be infinite for compact sets \(C\subset {\mathbb {R}}^n\) and \(w\in {\mathrm{Conv}({\mathbb {R}}^n)}\). An example is given by \(w\in {\mathrm{Conv}({\mathbb {R}}^n)}\) defined as
as we have
However, on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) we obtain a Radon measure, where a Borel measure M is called a Radon measure if \(M(C)<+\infty \) for every compact set \(C\subset {\mathbb {R}}^n\). This is the content of the following result, which is due to Aleksandrov [1] (see [21, Theorem 2.3] or [22, Theorem 1.1.13]). Let \({\mathcal {B}}({\mathbb {R}}^n)\) be the class of Borel sets in \({\mathbb {R}}^n\).
Lemma 4.1
Let \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). If \(B\in {\mathcal {B}}({\mathbb {R}}^n)\), then the set \(\partial v(B)\) is measurable. Moreover, \(\,\mathrm{MA}(v;\cdot ):{\mathcal {B}}({\mathbb {R}}^n)\rightarrow [0,\infty ]\), defined by
is a Radon measure on \({\mathbb {R}}^n\).
We will refer to \(\mathrm{MA}(v;\cdot )\) as the Monge–Ampère measure of v. The notion of Monge–Ampère measure is fundamental in the definition of weak or generalized solutions of the Monge–Ampère equation (see, for instance, [21, 22, 47]).
The following statement gathers properties of Monge–Ampère measures. Items (a) and (b) are due to Aleksandrov [1] (or see [21, Proposition 2.6 and Theorem A.31]) while the valuation property (c) was deduced by Alesker [4] from Błocki [9] (or see [15, Theorem 9.2]). Recall that for a sequence \(M_k\) of Radon measures in \({\mathbb {R}}^n\), we say that \(M_k\) converges weakly to a Radon measure M in \({\mathbb {R}}^n\) if
for every \(\beta \in C_c({\mathbb {R}}^n)\) (see, for instance, [20]).
Theorem 4.2
The following properties hold.
-
(a)
If \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(v\in C^2(V)\) on an open set \(V\subset {\mathbb {R}}^n\), then \(\mathrm{MA}(v;\cdot )\) is absolutely continuous on V with respect to n-dimensional Lebesgue measure and
$$\begin{aligned} \,\mathrm {d}\mathrm{MA}(v;x)=\det ({{\text {D}}}^2v(x))\,\mathrm {d}x \end{aligned}$$for \(x\in V\).
-
(b)
If \(v_j\) is a sequence in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) that is epi-convergent to \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), then the sequence of measures \(\mathrm{MA}(v_j;\cdot )\) converges weakly to \(\mathrm{MA}(v;\cdot )\).
-
(c)
For every \(v_1,v_2\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) such that \(v_1\wedge v_2\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\),
$$\begin{aligned} \mathrm{MA}(v_1;\cdot )+\mathrm{MA}(v_2;\cdot )=\mathrm{MA}(v_1\wedge v_2;\cdot )+\mathrm{MA}(v_1\vee v_2;\cdot ), \end{aligned}$$that is, \(\mathrm{MA}\) is a (measure-valued) valuation on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
Let \(\mathcal {M}({\mathbb {R}}^n)\) denote the space of Radon measures on \({\mathbb {R}}^n\). According to Theorem 4.2 (b), the map \(\mathrm{MA}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is continuous, when \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) is equipped with the topology induced by epi-convergence and \(\mathcal {M}({\mathbb {R}}^n)\) with the topology induced by weak convergence.
4.1 Mixed Monge–Ampère measures
We use polarization of the Monge–Ampère measure with respect to the standard addition of functions to obtain mixed Monge–Ampère measures. They were called mixed n-Hessian measures in [46] and were used, for example, in [39].
We say that a map \({{\,\mathrm{{\text {Z}}}\,}}:\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is symmetric, if the measure \({{\,\mathrm{{\text {Z}}}\,}}(v_1,\dots ,v_n;\cdot )\) is invariant with respect to every permutation of n-tuples of functions in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). For \(0\le j \le n\) and \(v,v_1, \dots , v_{n-j}\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), we write \({{\,\mathrm{{\text {Z}}}\,}}(v[j], v_1, \dots , v_{n-j}; \cdot )\) when the entry v is repeated j times.
Theorem 4.3
There exists a symmetric map \(\,\mathrm{MA}:\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\,\) which assigns to every n-tuple of functions \(v_1,\dots ,v_n\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) a Radon measure \(\mathrm{MA}(v_1,\dots ,v_n;\cdot )\) with the following properties.
-
(a)
For every \(m\in {\mathbb {N}}\), every m-tuple of functions \(v_1,\dots ,v_m\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), and \(\lambda _1,\dots ,\lambda _m\ge 0\),
$$\begin{aligned} \mathrm{MA}(\lambda _1 v_1+\dots +\lambda _m v_m;\cdot ) =\sum _{i_1,\dots ,i_n=1}^m \lambda _{i_1}\cdots \lambda _{i_n} \mathrm{MA}(v_{i_1},\dots , v_{i_n};\cdot ). \end{aligned}$$ -
(b)
For every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\),
$$\begin{aligned} \mathrm{MA}(v,\dots ,v;\cdot )=\mathrm{MA}(v;\cdot ). \end{aligned}$$ -
(c)
If \(v_1,\dots ,v_n\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(v_1,\dots ,v_n \in C^2(V)\) on an open set \(V\subset {\mathbb {R}}^n\), then \(\mathrm{MA}(v_1,\dots ,v_n;\cdot )\) is absolutely continuous on V with respect to n-dimensional Lebesgue measure and
$$\begin{aligned} \,\mathrm {d}\mathrm{MA}(v_1,\dots ,v_n;x)=\det ({{\text {D}}}^2v_1(x),\dots ,{{\text {D}}}^2v_n(x))\,\mathrm {d}x \end{aligned}$$for \(x\in V\).
-
(d)
The map \(\mathrm{MA}:\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is continuous, when \(\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\) is equipped with the product topology and every factor has the topology induced by epi-convergence, while \(\mathcal {M}({\mathbb {R}}^n)\) is equipped with the topology induced by weak convergence.
-
(e)
The map \(\mathrm{MA}:\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is dually epi-translation invariant with respect to every entry, that is,
$$\begin{aligned} \mathrm{MA}(v+\ell +\gamma ,v_1,\dots ,v_{n-1};\cdot )=\mathrm{MA}(v,v_1,\dots ,v_{n-1};\cdot ) \end{aligned}$$for every \(v,v_1,\dots ,v_{n-1}\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), every linear function \(\ell :{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and \(\gamma \in {\mathbb {R}}\).
-
(f)
The map \(\mathrm{MA}:\left( {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is additive and positively homogeneous of degree 1 with respect to every entry, that is,
$$\begin{aligned} \mathrm{MA}(\lambda v +\mu w,v_1,\dots ,v_{n-1};\cdot )&=\lambda \,\mathrm{MA}(v,v_1,\dots ,v_{n-1};\cdot )\nonumber \\&+\mu \,\mathrm{MA}(w,v_1,\dots ,v_{n-1}; \cdot ) \end{aligned}$$for every \(v,w,v_1,\dots ,v_{n-1}\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(\lambda ,\mu \ge 0\).
-
(g)
For \(0\le j\le n\) and \(v_1,\dots ,v_{n-j}\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), the map
$$\begin{aligned} v \mapsto \mathrm{MA}(v[j],v_1, \dots , v_{n-j}; \cdot ) \end{aligned}$$is a (measure-valued) valuation on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
Proof
For \(v_1,\dots ,v_n\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\) and \(B\in {\mathcal {B}}({\mathbb {R}}^n)\), we set
Note that this measure is non-negative and symmetric. If all functions are in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\), it verifies (a) by Theorem 4.2 (a) and by the fact that the mixed discriminant polarizes the determinant. Properties (b) and (f) for functions in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\) also follow from corresponding properties of the mixed discriminant. Property (e) for functions in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\) follows directly from the fact that adding an affine function to v does not change the Hessian matrix of v.
By (3.2), we have
for every \(v_1,\dots ,v_n\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\). This identity, combined with Theorem 4.2 (b) and the denseness of \(C^2({\mathbb {R}}^n)\) functions in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), shows that the definition of \(\mathrm{MA}\) extends continuously to \(({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})})^n\). Hence, we get properties (c) and (d). The extension inherits properties (a), (b), (e) and (f) by continuity. Property (g) follows from (4.1) and Theorem 4.2 (c). \(\square \)
Concerning (c), note that we use different notation for the Lebesgue integral over the mixed discriminant, which we only consider for functions that are of class \(C^2\) almost everywhere, and the mixed Monge–Ampère measure. This distinction is not always made in the literature.
As a consequence of Theorem 4.3 we obtain the following result, which for the special case \(j=n\) was previously established in [14, Proposition 19].
Proposition 4.4
Let \(\beta \in C_c({\mathbb {R}}^n)\) and \(0\le j \le n\). If \(v_{1},\dots ,v_{n-j}\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), then the map \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) defined by
is a continuous, dually epi-translation invariant valuation that is homogeneous of degree j.
Proof
Note that the integral in (4.2) is well-defined and finite as \(\beta \in C_c({\mathbb {R}}^n)\) and mixed Monge–Ampère measures are Radon measures. Continuity follows from the weak continuity of mixed Monge–Ampère measures. The invariance, homogeneity and valuation properties are consequences of items (e), (f) and (g) of Theorem 4.3, respectively. \(\square \)
We remark that valuations defined in a way similar to (4.2) have been considered by Alesker [4] and by Knoerr [29].
4.2 Hessian measures as a special case
For a function \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\) and \(0\le j\le n\), the jth Hessian measure \(\Phi _{j}(v;B)\) is defined for \(B\in {\mathcal {B}}({\mathbb {R}}^n)\) as
Trudinger and Wang [44, 45] showed that \(\Phi _{j}(v;\cdot )\) can be extended to a Radon measure on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). It coincides up to the factor \(\left( {\begin{array}{c}n\\ j\end{array}}\right) \) with
Indeed, if \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\), then by Theorem 4.3 (c) and (3.4),
We obtain the conclusion using the denseness of smooth functions.
4.3 A special family of mixed Monge–Ampère measures
We introduce the following family of mixed Monge–Ampère measures. For \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(0\le j \le n\), we set
By construction, this is a Radon measure on \({\mathbb {R}}^n\).
It follows from Theorem 4.3 (a) that the mixed Monge–Ampère measures (4.3) can also be obtained as coefficients of the following Steiner formula,
for \(B\in {\mathcal {B}}({\mathbb {R}}^n)\) and \(r\ge 0\).
We derive some of the properties of the measures \(\mathrm{MA}_{j}(v;\cdot )\) for \(0\le j\le n\). The subdifferential of \(h_{B^n}\) can be explicitly described as
Combining this with the definition of Monge–Ampère measure, we see that
where \(\delta _0\) is the Dirac measure at 0. Indeed, if \(B\in {\mathcal {B}}({\mathbb {R}}^n)\) does not contain the origin, then we have \(\partial h_{B^n}(B)\subset \mathbb {S}^{n-1}\), so that
On the other hand, if \(0\in B\), then \(\partial h_{B^n}(B)=B^n\). Note that
for \(x\ne 0\), where \(y\otimes z\) denotes the tensor product of \(y,z\in {\mathbb {R}}^n\).
Theorem 4.5
Let \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). The following properties hold.
-
(a)
For \(B\in {\mathcal {B}}({\mathbb {R}}^n)\),
$$\begin{aligned} \mathrm{MA}_{0}(v;B)=\kappa _n\delta _0(B). \end{aligned}$$In particular, \(\,\mathrm{MA}_{0}(v;\cdot )\) is independent of v.
-
(b)
For \(B\in {\mathcal {B}}({\mathbb {R}}^n)\),
$$\begin{aligned} \mathrm{MA}_{n}(v;B)=\vert {\partial v(B)}\vert =\mathrm{MA}(v;B). \end{aligned}$$ -
(c)
For \(\,0\le j\le n\),
$$\begin{aligned} \mathrm{MA}_{j}(v;\{0\})=\frac{\kappa _{n-j}}{\left( {\begin{array}{c}n\\ j\end{array}}\right) }\, V_{j}(\partial v(0)). \end{aligned}$$ -
(d)
If \(v\in C^2(V)\) with \(V\subset {\mathbb {R}}^n\) open and \(1\le j\le n\), then \(\mathrm{MA}_{j}(v;\cdot )\) is absolutely continuous on \(V\backslash \{0\}\) with respect to n-dimensional Lebesgue measure and
$$\begin{aligned} \,\mathrm {d}\mathrm{MA}_{j}(v;x)=\det ({{\text {D}}}^2v(x)[j],{{\text {D}}}^2h_{B^n}(x) [n-j])\,\mathrm {d}x \end{aligned}$$for \(x\in V\) with \(x\ne 0\).
-
(e)
For \(\,0\le j\le n\), the map \(\mathrm{MA}_{j}: {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is a continuous valuation.
Proof
Item (a) follows from (4.6). Item (b) follows from (4.3) with \(j=n\) and the definition of \(\mathrm{MA}(v;\cdot )\). Item (d) is a consequence of Theorem 4.3 (d) and (g) while item (c) follows from Theorem 4.3 (c) combined with (4.7) and (3.1).
It remains to show (c). By (4.5) we obtain that for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(r\ge 0\),
where we use that the subdifferential of the sum of two convex functions is the Minkowski sum of their subdifferentials (see, for example, [41, Theorem 23.8]). Hence, according to the Steiner formula (1.1),
which combined with the definition of Monge–Ampère measure and (4.4) concludes the proof. \(\square \)
5 Conjugate Monge–Ampère and conjugate mixed Monge–Ampère measures
First, we define the conjugate Monge–Ampère measure for super-coercive convex functions, using the construction of Monge–Ampère measures on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and a duality argument. Let \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). For \(B\in {\mathcal {B}}({\mathbb {R}}^n)\), we set
Note that Lemma 4.1 implies that \(\mathrm{MA}^{\!*}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is well-defined and that \(\mathrm{MA}^{\!*}(u;\cdot )\) is a Radon measure for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), as \(u^*\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) in this case. We refer to \(\mathrm{MA}^{\!*}(u; \cdot )\) as the conjugate Monge–Ampère measure of u. It is the push-forward through \(\nabla u\) of n-dimensional Lebesgue measure restricted to the domain of u and we have included a proof of this known fact as item (a) of the following result. In the following, for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), we use the relation
for \(x\in {\mathbb {R}}^n\) such that \(u\in C_+^2(U)\) in a neighborhood U of x (see [42, p. 605]).
Theorem 5.1
The following properties hold.
-
(a)
If \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), then
$$\begin{aligned} \int _{{\mathbb {R}}^n}\beta (y)\,\mathrm {d}\mathrm{MA}^{\!*}(u;y)=\int _{{\text {dom}}u}\beta (\nabla u(x)) \,\mathrm {d}x \end{aligned}$$for every \(\beta \in C_c({\mathbb {R}}^n)\).
-
(b)
If \(u_j\) is a sequence in \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) that is epi-convergent to \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), then the sequence of measures \(\mathrm{MA}^{\!*}(u_j;\cdot )\) converges weakly to \(\mathrm{MA}^{\!*}(u;\cdot )\).
-
(c)
For every \(u_1,u_2\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) such that \(u_1\vee u_2\) and \(u_1\wedge u_2\) are in \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\),
$$\begin{aligned} \mathrm{MA}^{\!*}(u_1;\cdot )+\mathrm{MA}^{\!*}(u_2;\cdot )=\mathrm{MA}^{\!*}(u_1\wedge u_2;\cdot )+\mathrm{MA}^{\!*}(u_1\vee u_2;\cdot ), \end{aligned}$$that is, \(\mathrm{MA}^{\!*}\) is a (measure-valued) valuation on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\).
Proof
Properties (b) and (c) are consequences of properties (b) and (c) in Theorem 4.2, respectively, and of Proposition 3.1.
Concerning property (a), observe that if \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C^2_+({\mathbb {R}}^n)\), then \(u^*\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2_+({\mathbb {R}}^n)\) (by Proposition 3.1). By Theorem 4.2 (a), setting \(v:=u^*\), we obtain
for \(\beta \in C_c({\mathbb {R}}^n)\). Here we used the change of variable \(y=\nabla u(x)\) and (5.2). The statement now follows from property (b) combined with the fact that the functional \(u\mapsto \int _{{\mathbb {R}}^n}\beta (\nabla u(x))\,\mathrm {d}x\) is continuous on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) (see (1.4)). \(\square \)
5.1 Conjugate mixed Monge–Ampère measures
We use polarization of the conjugate Monge–Ampère measure with respect to infimal convolution to define conjugate mixed Monge–Ampère measures. The following result is easily obtained from Theorem 4.3.
Theorem 5.2
There exists a symmetric map \(\,\mathrm{MA}^{\!*}:\left( {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\,\) which assigns to every n-tuple of functions \(u_1,\dots ,u_n\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) a Radon measure \(\mathrm{MA}^{\!*}(u_1,\dots ,u_n;\cdot )\) with the following properties.
-
(a)
For every \(m\in {\mathbb {N}}\), every m-tuple of functions \(u_1,\dots ,u_m\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(\lambda _1,\dots ,\lambda _m\ge 0\),
-
(b)
For every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\),
$$\begin{aligned} \mathrm{MA}^{\!*}(u,\dots ,u;\cdot )=\mathrm{MA}^{\!*}(u;\cdot ). \end{aligned}$$ -
(c)
If \(\,u_1,\dots ,u_n\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(u_1^*,\dots ,u_n^*\in C^2(V)\) on an open set \(V\subset {\mathbb {R}}^n\), then the measure \(\mathrm{MA}^{\!*}(u_1,\dots ,u_n;\cdot )\) is absolutely continuous on V with respect to n-dimensional Lebesgue measure and
$$\begin{aligned} \,\mathrm {d}\mathrm{MA}^{\!*}(u_1,\dots ,u_n;x)=\det ({{\text {D}}}^2u_1^*(x),\dots ,{{\text {D}}}^2u_n^*(x))\,\mathrm {d}x\end{aligned}$$for \(x\in V\).
-
(d)
The map \(\mathrm{MA}^{\!*}:\left( {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is continuous, when \(\left( {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\right) ^n\) is equipped with the product topology and every factor has the topology induced by epi-convergence, while \(\mathcal {M}({\mathbb {R}}^n)\) is equipped with the topology induced by weak convergence.
-
(e)
The map \(\mathrm{MA}^{\!*}:\left( {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is epi-translation invariant with respect to every entry, that is,
$$\begin{aligned} \mathrm{MA}^{\!*}(u\circ \tau ^{-1} +\gamma ,u_1,\dots ,u_{n-1};\cdot )=\mathrm{MA}^{\!*}(u,u_1,\dots ,u_{n-1};\cdot ) \end{aligned}$$for every \(u,u_1,\dots ,u_{n-1}\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), every translation \(\tau :{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) and \(\gamma \in {\mathbb {R}}\).
-
(f)
The map \(\mathrm{MA}^{\!*}:\left( {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\right) ^n\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is epi-additive and epi-homogeneous of degree 1 with respect to every entry, that is,
for every \(u,w,u_1,\dots ,u_{n-1}\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and for every \(\lambda ,\mu \ge 0\).
-
(g)
For \(\,0\le j\le n\) and \(u_1, \dots , u_{n-j}\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), the map
$$\begin{aligned} u\mapsto \mathrm{MA}^{\!*}(u[j],u_1, \dots ,u_{n-j};\cdot ) \end{aligned}$$is a (measure-valued) valuation on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\).
Here, for (a) and (f), we extend the definition of epi-multiplication to 
for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\).
The dual version of Proposition 4.4 is the following result.
Proposition 5.3
Let \(\beta \in C_c({\mathbb {R}}^n)\) and \(0\le j\le n\). If \(u_{1},\dots ,u_{n-j}\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), then the map \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow {\mathbb {R}}\), defined by
is a continuous, epi-translation invariant valuation, that is epi-homogeneous of degree j.
5.2 Connections to Hessian measures
For \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C_+^2({\mathbb {R}}^n)\) and \(0\le j\le n\), define the Hessian measure \(\Psi _{j}(u; \cdot )\) as the push-forward through \(\nabla u\) of the Hessian measure \(\Phi _{n-j}(u;\cdot )\) of u, that is,
for every Borel function \(\beta :{\mathbb {R}}^n\rightarrow [0,\infty )\). We remark that the measure \(\Psi _{j}(u;\cdot )\) can be defined for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and is a marginal of a generalized Hessian measure (see [15]). Moreover, for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\cap C_+^2({\mathbb {R}}^n)\) we obtain from (5.1) and Theorem 4.3 (c) that
where for the last step we used (5.2) and (3.5). Hence, the measure
coincides up to the factor \(\left( {\begin{array}{c}n\\ j\end{array}}\right) \) with \(\Psi _{j}(u;\cdot )\) for \(u \in C_+^2({\mathbb {R}}^n)\). The corresponding statement holds for general \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) by the denseness of smooth functions and the weak continuity of Hessian and conjugate mixed Monge–Ampère measures.
5.3 A special family of conjugate mixed Monge–Ampère measures
We introduce the following family of conjugate mixed Monge–Ampère measures. For \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(0\le j \le n\), we set
By construction, this is a Radon measure on \({\mathbb {R}}^n\). A consequence of this definition is that
for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). It follows from Theorem 5.2 (a) that this family of conjugate mixed Monge–Ampère measures can also be obtained as coefficients of the following Steiner formula,
for \(B\in {\mathcal {B}}({\mathbb {R}}^n)\) and \(r\ge 0\).
The next result describes properties of this family of conjugate Monge–Ampère measures.
Theorem 5.4
Let \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). The following statements hold.
-
(a)
For every \(B\in {\mathcal {B}}({\mathbb {R}}^n)\),
$$\begin{aligned} \mathrm{MA}^{*}_{0}(u;B)=\kappa _n\delta _0(B). \end{aligned}$$In particular, \(\,\mathrm{MA}^{*}_{0}(u;\cdot )\) is independent of u.
-
(b)
For every \(B\in {\mathcal {B}}({\mathbb {R}}^n)\),
$$\begin{aligned} \mathrm{MA}^{*}_{n}(u;B)=\vert {\partial u^*(B)}\vert =\mathrm{MA}^{\!*}(u;B). \end{aligned}$$ -
(c)
For \(\,0\le j\le n\),
$$\begin{aligned} \mathrm{MA}^{*}_{j}(u;\{0\})=\frac{\kappa _{n-j}}{\left( {\begin{array}{c}n\\ j\end{array}}\right) }\, V_{j}({\text {argmin}}u). \end{aligned}$$ -
(d)
If \(u^*\in C^2_+(V)\) with \(V\subset {\mathbb {R}}^n\) open and \(1\le j\le n\), then \(\mathrm{MA}^{*}_{j}(u;\cdot )\) is absolutely continuous on V with respect to n-dimensional Lebesgue measure and
$$\begin{aligned} \,\mathrm {d}\mathrm{MA}^{*}_{j}(u;x)=\det ({{\text {D}}}^2u^*(x)[j],{{\text {D}}}^2h_{B^n}(x) [n-j])\,\mathrm {d}x \end{aligned}$$for \(x\in V\) with \(x\ne 0\).
-
(e)
For \(\,0\le j\le n\), the map \(\mathrm{MA}^{*}_{j}:{\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\rightarrow \mathcal {M}({\mathbb {R}}^n)\) is a continuous valuation.
Proof
The statements follow from the corresponding statements in Theorem 4.5 by duality. For (c), we use that \(\partial u^*(0)={\text {argmin}}u\) (see [42, Theorem 11.8]). \(\square \)
In the following, for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\), we will use the fact that \(y\in \partial u(x)\) if and only if \(x\in \partial u^*(y)\) (see, for example, [41, Theorem 23.5]). Combined with (5.2), it implies that \(u\in C_+^2(U)\) for some open set \(U\subset {\mathbb {R}}^n\) if and only if \(u^*\in C_+^2(V)\) with \(V:=\{\nabla u(x) :x\in U\}\). In particular, we obtain that the set V is open and \(\nabla u:U\rightarrow V\) is a bijection.
For the following result, we recall that \(\tau _i(u,x)\) is the ith elementary symmetric function of the principal curvatures of the sublevel set of u passing through x for \(x\notin {\text {argmin}}u\).
Theorem 5.5
Let \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(1\le j\le n-1\). If \(u\in C_+^2(U)\) for an open set \(U\subset {\mathbb {R}}^n\) and \(V:=\{\nabla u(x) :x\in U\}\), then
for every Borel set \(B\subset V\backslash \{0\}\). Equivalently,
for every \(\beta \in C_c(V)\) .
For the proof we need the following result.
Lemma 5.6
Let \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) be such that \(u\in C_+^2(U)\) for an open set \(U \subset {\mathbb {R}}^n\) and let \(r>0\). If \(T_r:U\backslash {\text {argmin}}u\rightarrow {\mathbb {R}}^n\) is defined by
then, for the Jacobi matrix \({{\text {D}}}T_r\), we have
for every \(x\in U\backslash {\text {argmin}}u\).
Proof
Let \(x\in U\backslash {\text {argmin}}u\). Clearly,
where \(N(x)=(N_1(x),\dots ,N_n(x))\) is defined as
Let \(t:=u(x)\). We choose a coordinate system such that
where \(\nu _t(x)\) denotes the outer unit normal to \(\{u\le t\}\) at x and \(\lambda =|\nabla u(x)|>0\). We may also assume that, for \(1\le j\le n-1\), the vector \(e_j\) is a direction of principal curvature for \(\partial \{u\le t\}\) at x with corresponding principal curvature \(\kappa _j(u,x)\). As N is an extension of \(\nu _t\), we obtain
On the other hand, using (5.7), we obtain
for \(1\le j\le n\). Here, for the last equality, we used that \(\frac{\partial u(x)}{\partial x_i}=0\) for all \(1\le i \le n-1\) because of the choice of our coordinate system. Therefore,
and
which implies the representation formula. \(\square \)
Proof of Theorem 5.5
Formula (5.6) directly follows from (5.5). So, we have to prove (5.5).
Let \(B\subset V\backslash \{0\}\) be a Borel set. For \(r>0\), let \(T_r:U\backslash {\text {argmin}}u\) be the map defined in Lemma 5.6. Note that \(U\backslash {\text {argmin}}u=\nabla u^{-1}(V\backslash \{0\})\). We have
where we have used Lemma 5.6. On the other hand, by the definition of the Monge–Ampère measure and of the conjugate Monge–Ampère measure and (5.4), we have
The conclusion follows from comparing coefficients. \(\square \)
6 Connecting \(\mathrm{MA}_{j}(v; \cdot )\) and Hessian measures
The purpose of this section is to prove Proposition 6.7, which shows how integrals of radially symmetric functions with respect to Hessian measures can be written in terms of integrals with respect to the new family of mixed Monge–Ampère measures. This result is essential for our new proof of the existence of functional intrinsic volumes, Theorems 1.2 and 2.1, as well as for the proof of the new representations, Theorems 1.6 and 2.5.
6.1 Reilly-type lemmas
We will need the following result by Reilly [40, Proposition 2.1] (or see [46, (2.10)]).
Lemma 6.1
(Reilly) If \(v_1,\ldots ,v_{n-1}\in C^3({\mathbb {R}}^n)\) and \(1\le j\le n\), then
for every \(x\in {\mathbb {R}}^n\).
The following result shows that
with \(v_0,\ldots ,v_n\in C^2({\mathbb {R}}^n)\) is symmetric in its entries if at least one the functions has compact support. We remark that this corresponds to the symmetry of mixed volumes in the following representation,
for sufficiently smooth \(K_1,\ldots ,K_n\in {\mathcal {K}}^n\), where \({\tilde{{\text {D}}}}^2h_K(y)\) is the restriction of \({{\text {D}}}^2h_K\) to the tangent space of \({\mathbb {S}^{n-1}}\) at \(y\in {\mathbb {S}^{n-1}}\) (see for example equations (2.68) and (5.19) in [43]).
Lemma 6.2
If \(v_0,\ldots ,v_n\in C^2({\mathbb {R}}^n)\) are such that at least one of the functions has compact support, then
Proof
Assume first that \(v_0,\ldots ,v_n\in C^3({\mathbb {R}}^n)\). In this case, \(D_{ij}({{\text {D}}}^2v_2(x),\ldots ,{{\text {D}}}^2v_n(x))\) is differentiable and therefore
for \(1\le i\le n\) and \(x\in {\mathbb {R}}^n\). Summation over i combined with Lemma 6.1 now gives
for \(x\in {\mathbb {R}}^n\). By the definition of \(D_{ij}\) and using that at least one of the functions \(v_0,\ldots ,v_n\) has compact support, we now obtain from the divergence theorem that
Since the last expression is symmetric in \(v_0\) and \(v_n\), we may exchange the two functions. This completes the proof under the additional assumption that all functions are in \(C^3({\mathbb {R}}^n)\).
It remains to show that the result holds true on \(C^2({\mathbb {R}}^n)\). By the multilinearity of the mixed discriminant combined with the assumption that one of the functions \(v_0,\ldots ,v_n\) has compact support, there exists \(r>0\) such that the integrands in (6.1) vanish outside of \(r B^n\). The result now easily follows by a standard approximation argument combined with the dominated convergence theorem. \(\square \)
A further consequence of Lemma 6.1 is the following result.
Lemma 6.3
Let \(v_1,\ldots ,v_{n-1}\in C^2({\mathbb {R}}^n)\) and let \(F:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) be a continuously differentiable vector field. If F has compact support, then
Proof
Assume first that \(v_1,\ldots ,v_{n-1}\in C^3({\mathbb {R}}^n)\). Using the definition of \(D_{ij}\) and that F has compact support, we obtain from the divergence theorem that
and the statement follows from Lemma 6.1.
As in the proof of Lemma 6.2, the general case follows from the fact that F has compact support combined with a standard approximation argument and the dominated convergence theorem. \(\square \)
6.2 Applications to mixed Monge–Ampère integrals
In the following we consider special integrals of mixed discriminants where the support function of the unit ball \(B^n\) appears repeatedly.
First, we show that such integrals are well-defined. Recall that \({{\text {D}}}^2h_{B^n}(x)\) exists for every \(x\ne 0\). We remark that throughout this subsection, Lebesgue integrals with respect to the standard Lebesgue measure on \({\mathbb {R}}^n\) are considered.
Lemma 6.4
Let \(1\le k \le n\). If \(\zeta \in C_b((0,\infty ))\) is such that \(\lim _{r\rightarrow 0^+} r^{k-1}\zeta (r)\) exists and is finite, then the integral
is well-defined and finite for every \(v_1,\ldots ,v_k\in C^2({\mathbb {R}}^n)\).
Proof
Fix \(v_1,\ldots ,v_k \in C^2({\mathbb {R}}^n)\) and let \(w\in C({\mathbb {R}}^n\backslash \{0\})\) be defined by
for \(x\in {\mathbb {R}}^n\backslash \{0\}\). By the multilinearity of the mixed discriminant and (4.7) the function w is bounded on \(B^n\backslash \{0\}\). Using polar coordinates, we obtain
where \(\mathcal {H}^{n-1}\) denotes the \((n-1)\)-dimensional Hausdorff measure. The result now follows from our assumptions on \(\zeta \) together with the fact that w is bounded on \(B^n\backslash \{0\}\). \(\square \)
The following result shows how replacing \(h_{B^n}\) by \(\tfrac{1}{2} h_{B^n}^2\) once in the mixed discriminant corresponds to taking an integral transform of the density function.
Lemma 6.5
Let \(1\le k \le n-1\) and let \(\varepsilon >0\). If \(v_1,\ldots ,v_k\in C^2({\mathbb {R}}^n)\) and \({{\text {D}}}^2v_1(x)=0\) for every \(x\in \varepsilon B^n\), then
for every \(\psi \in C_b^2((0,\infty ))\), where \(\rho \in C_b((0,\infty ))\) is given for \(s>0\) by
Proof
Observe that our assumptions on \(v_1\) imply that the mixed discriminants in both integrals vanish on \(\varepsilon B^n\). Since the support of \(\psi \) is bounded, this implies that both integrals are well-defined and finite.
Let \(\xi (t)=\int _t^{\infty } \frac{\psi (s)}{s^2} \,\mathrm {d}s\) for \(t>0\). Since \(\psi \in C_b^2((0,\infty ))\) we have \(\xi \in C_b^3((0,\infty ))\) and furthermore \(\psi (t)=-\xi '(t) t^2\) as well as \(\frac{\psi (t)}{t}=-\xi '(t)t\) for \(t>0\). Thus, we need to show that
for every \(v_1,\ldots ,v_{k}\in C^2({\mathbb {R}}^n)\). Since the mixed discriminants in both integrals vanish on \(\varepsilon B^n\), we can replace \(h_{B^n}\) as well as \(x\mapsto \xi '(|x|)|x|^2\) and \(x\mapsto \xi '(|x|)|x|+\xi (|x|)\) by suitable functions in \(C^2({\mathbb {R}}^n)\) without changing the values of the integrals. Thus after applying Lemma 6.2 and changing back to the original functions, we obtain that (6.2) is equivalent to
Using the multilinearity of mixed discriminants, it suffices to show that
vanishes. Since we have
and
we obtain
for every \(x\in {\mathbb {R}}^n\backslash \{0\}\), where \({{\text {D}}}(\psi '(|x|)x)\) denotes the Jacobian of the vector field \(x\mapsto \psi '(|x|)x\). The result now follows from Lemma 6.3 and the definition of \(D_{ij}\), where we have used again that we may replace the integrands in a neighborhood of the origin. \(\square \)
In the next two statements we remove the regularity assumptions of the last result and treat the case where the support function of the unit ball \(B^n\) is replaced multiple times.
Proposition 6.6
If \(\,1\le k \le n-1\) and \(\psi \in D_{n-k+1}^{n}\), then
for every \(v_1,\ldots ,v_k\in C^2({\mathbb {R}}^n)\), where \({\mathcal {R}}^{-1}\) was defined in Sect. 3.3.
Proof
Since \(\psi \in D_{n-k+1}^{n}\), there exists \(\gamma >0\) such that \(\psi (t)=0\) for every \(t\ge \gamma \). Note that by Lemma 3.2 this implies that \({\mathcal {R}}^{-1} \psi (t)=0\) for every \(t\ge \gamma \). We will assume first that there exists \(\varepsilon >0\) such that \({{\text {D}}}^2v_1(x)=0\) for every \(x\in \varepsilon B^n\).
Let \(\psi _\varepsilon \in C_b((0,\infty ))\) be such that \(\psi _\varepsilon \equiv \psi \) on \([\varepsilon ,\infty )\) and \(\psi _\varepsilon \equiv 0\) on \((0,\varepsilon /2]\). Observe that this implies that \({\mathcal {R}}^{-1}\psi _\varepsilon \equiv {\mathcal {R}}^{-1}\psi \) on \([\varepsilon ,\infty )\). For \(\delta >0\) we can find \(\psi _{\varepsilon ,\delta }\in C_b^2((0,\infty ))\) such that \(\psi _{\varepsilon ,\delta }\equiv 0\) on \((0,\varepsilon /2]\cup [\gamma +\delta ,\infty )\) and such that \(\psi _{\varepsilon ,\delta }\rightarrow \psi _\varepsilon \) uniformly on \((\varepsilon /2,\gamma +\delta )\) (and thus on \((0,\infty )\)) as \(\delta \rightarrow 0^+\). By the properties of \(\psi _\varepsilon \) this also implies uniform convergence of \({\mathcal {R}}^{-1}\psi _{\varepsilon ,\delta }\) to \({\mathcal {R}}^{-1}\psi _\varepsilon \) on \((0,\infty )\) as \(\delta \rightarrow 0^+\). Using that \({{\text {D}}}^2v_1\equiv 0\) on \(\varepsilon B^n\), Lemma 6.5, as well as the fact that the integrands in each of the following integrals are continuous and have compact support, we now have
which completes the proof under the additional assumptions on \(v_1\).
For general \(v_1\in C^2({\mathbb {R}}^n)\), observe that without loss of generality we may assume that \(v_1(0)=0\) and \(\nabla v_1(0)=0\). Thus, there exist \(\beta ,\varepsilon _0>0\) such that \(|v_1(x)|\le \beta |x|^2\) and \(|\nabla v_1(x)|\le \beta |x|\) for every \(x\in 2\varepsilon _0 B^n\). Let \(\varphi \in C^2([0,\infty ))\) be such that \(\varphi (t)=0\) for \(t\in [0,1]\) and \(\varphi (t)=1\) for \(t\in [2,\infty )\). For \(\varepsilon \in (0,\varepsilon _0)\), set \(v_{1,\varepsilon }(x):=v_1(x)\, \varphi (|x|/\varepsilon )\) for \(x\in {\mathbb {R}}^n\). We now have \({{\text {D}}}^2v_{1,\varepsilon }(x)= 0\) for every \(x\in \varepsilon B^n\) and our assumptions on \(v_1\) together with the fact that \(\varphi \) is constant on \([2,\infty )\) imply that \({{\text {D}}}^2v_{1,\varepsilon }\) is uniformly bounded on \(\gamma B^n\) for every \(\varepsilon \in (0,\varepsilon _0)\). Moreover, \({{\text {D}}}^2v_{1,\varepsilon }\rightarrow {{\text {D}}}^2v_1\) pointwise on \({\mathbb {R}}^n\) as \(\varepsilon \rightarrow 0^+\). Since \(\psi \in D_{n-k+1}^{n}\) the limit \(\lim _{t\rightarrow 0^+} t^{k-1}\psi (t)\) exists and is finite. By Lemma 3.2 and since \(D_{n-k+1}^{n}=D_{n-k}^{n-1}\), we have \({\mathcal {R}}^{-1}\psi \in D_{n-k}^{n}\) and thus also \(\lim _{t\rightarrow 0^+} t^{k}{\mathcal {R}}^{-1}\psi (t)\) exists and is finite. Hence, by the first part of the proof and Lemma 6.4 combined with the dominated convergence theorem we now obtain
which concludes the proof. \(\square \)
Proposition 6.7
If \(\,1\le j \le n-1\) and \(\zeta \in D_{j}^{n}\), then
for every \(v\in C^2({\mathbb {R}}^n)\).
Proof
Let \(1\le j \le n-1\) and \(\zeta \in D_{j}^{n}\) be given. We claim that
for every \(k\in {\mathbb {N}}\) such that \(j\le k \le n-1\) and every \(v\in C^2({\mathbb {R}}^n)\). Indeed, as \(\zeta \in D_{j}^{n}\) it follows from Lemma 3.2 that \({\mathcal {R}}^{n-k}\zeta \in D_{j}^{k}\). Since \(D_{j}^{k}=D_{n-k+j}^{n}\) and \(D_{n-k+j}^{n}\subseteq D_{n-k+1}^{n}\), the claim now follows from Proposition 6.6.
Applying (6.3) recursively \((n-j)\) times (for each possible value of k), we obtain that
for every \(v\in C^2({\mathbb {R}}^n)\). The statement now follows from (3.4). \(\square \)
7 Proofs of the main results
In this section, we present a new proof of the existence of functional intrinsic volumes, Theorems 1.2 and 2.1. Moreover, we prove our main results: the new representations of functional intrinsic volumes, Theorems 1.6 and 2.5, as well as the Steiner formulas, Theorems 1.4 and 2.3.
By the duality relations between valuations on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), it is enough to prove the results for valuations on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), that is, to prove Theorems 2.1, 2.3 and 2.5. Theorems 1.2 and 1.4 are immediate consequences of their counterparts on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) while Theorem 1.6 follows from Theorem 2.5 combined with Theorem 5.5.
7.1 New proof of Theorem 2.1
The statement is trivial for \(j=0\) and follows from Proposition 4.4 for \(j=n\). So, let \(1\le j \le n-1\) and \(\zeta \in D_{j}^{n}\). We set \(\alpha :=\left( {\begin{array}{c}n\\ j\end{array}}\right) {\mathcal {R}}^{n-j} \zeta \) and note that, by Lemma 3.2, we have \(\alpha \in D_{n}^{n}\). We define \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) by
By Proposition 4.4, the definition of \(\mathrm{MA}_{j}(v;\cdot )\), and (4.3), the functional \({{\,\mathrm{{\text {Z}}}\,}}\) is a continuous and dually epi-translation invariant valuation on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). It is easy to see that \({{\,\mathrm{{\text {Z}}}\,}}\) is rotation invariant. For \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\), it follows from Theorem 4.5 (c) and (d) that
and thus, by Proposition 6.7, the valuation \({{\,\mathrm{{\text {Z}}}\,}}\) satisfies (2.1) for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\).
We conclude that \({{\,\mathrm{{\text {Z}}}\,}}\) has the required properties and remark that it is uniquely determined by (2.1), since \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C_+^2({\mathbb {R}}^n)\) is dense in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
7.2 Proof of Theorem 2.5
For \(j=n\) the statement trivially follows from Proposition 4.4 and Theorem 4.5 (d). Next, consider the case \(j=0\) and let \(\zeta \in D_{0}^{n}\) and \(\alpha \in C_c([0,\infty ))\) be as in the statement of the theorem. Using polar coordinates and (4.6) we now have
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), which proves the statement for \(j=0\). Finally, let \(1\le j \le n-1\) and \(\zeta \in D_{j}^{n}\). For \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\), it follows from (2.1), which was established in the new proof of Theorem 2.1 for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\), and Proposition 6.7 that
Combining this with Theorem 4.5 (c) and (d), we obtain
The statement now follows from Proposition 4.4 and the fact that \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C^2({\mathbb {R}}^n)\) is dense in \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
7.3 Proof of Theorem 2.3
Let \(\zeta \in D_{n}^{n}\) be given and for \(0\le j\le n\), let \(\zeta _j\in D_{j}^{n}\) be defined as in (2.2). By Theorem 2.5 and (4.4) we have
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(r>0\). Using Theorem 2.5 again and Lemma 3.2, we obtain that
for every \(0\le j \le n\) and \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\), which concludes the proof.
8 Additional results and applications
In this section we prove additional results and derive further applications. Section 8.1 contains a second proof of the functional Steiner formula, Theorem 1.4, which uses the Hadwiger Theorem on convex functions, Theorem 1.3. In the subsequent subsection we use the properties of mixed Monge–Ampère measures to obtain a new representation of functional intrinsic volumes. In Sect. 8.3, we show how the classical Steiner formula (1.1) can be retrieved from our new functional version. The final subsection partly answers the question, which functions playing the role of the unit ball give rise to all functional intrinsic volumes in a Steiner-type formula.
8.1 Alternate proof of Theorem 1.4
Our approach for this proof follows the proof of the classical Steiner formula from [27, Theorem 9.2.3] and uses Theorem 1.3. We remark that multinomiality with respect to infimal convolution of continuous, epi-translation invariant valuations on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) was established by the authors in [14]. For polynomial expansions for a different functional analog of the volume on convex functions, see Milman and Rotem [36].
Let \(\zeta \in D_{n}^{n}\) be given. It is easy to see that \(u\mapsto {\text {V}}_{n,\zeta }(u\mathbin {\Box }\mathbf{I}_{B^n})\) defines a continuous, epi-translation and rotation invariant valuation on \({\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). Thus, by Theorem 1.3 there exist functions \(\tilde{\zeta }_j\in D_{j}^{n}\) for \(0\le j \le n\) such that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\). Using the epi-homogeneity of functional intrinsic volumes, we now have
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(r>0\).
In order to determine the functions \(\tilde{\zeta }_j\) for \(0\le j \le n\), we evaluate the last expression for \(u=u_t\) with \(t> 0\), where \(u_t(x):= t \vert x\vert + \mathbf{I}_{B^n}(x)\) for \(x\in {\mathbb {R}}^n\). Since
a simple calculation shows that
for every \(r>0\) and \(t> 0\). A comparison of coefficients combined with Lemma 3.3 shows that \({\mathcal {R}}^{n-j}\tilde{\zeta }_j(t)=\zeta (t)\) for every \(t> 0\) and \(1\le j \le n\). Thus, by Lemma 3.2, we get \(\tilde{\zeta }_j={\mathcal {R}}^{-(n-j)} \zeta \) for every \(1\le j \le n\).
For \(j=0\), observe that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(\xi \in D_{0}^{n}\). Thus, our calculations combined with Lemma 3.2 and the definition of \(D_{0}^{n}\) show that
for every \(t>0\). Since \({\text {V}}_{0,\tilde{\zeta }_0}(u)\) is independent of \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and only depends on \({\mathcal {R}}^n \tilde{\zeta }_0(0)\), it easily follows from Lemma 3.2 that we may choose \(\tilde{\zeta }_0 = {\mathcal {R}}^{-n} \zeta \).
The result now follows by setting \(\zeta _j=\frac{1}{\kappa _{n-j}} \tilde{\zeta }_j = \frac{1}{\kappa _{n-j}} {\mathcal {R}}^{-(n-j)} \zeta \) and observing that
for every \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(0\le j \le n\).
8.2 Representation formulas for functional intrinsic volumes
Let \(1\le j \le n\). By Theorem 2.5,
for \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) and \(\zeta \in D_{j}^{n}\), where \(\alpha : = \left( {\begin{array}{c}n\\ j\end{array}}\right) {\mathcal {R}}^{n-j}\zeta \). If, in addition, \(v\in C^2({\mathbb {R}}^n\backslash \{0\})\), then Theorem 4.5 (c) and (d) imply that
Correspondingly, by Theorem 1.6, we obtain that
for \(u\in {\mathrm{Conv}_{\mathrm{sc}}({\mathbb {R}}^n)}\) and \(\zeta \in D_{j}^{n}\), where \(\alpha \) is defined as before. If, in addition, \(u\in C_+^2({\mathbb {R}}^n\backslash {\text {argmin}}u)\), then Theorem 5.4 (c) and (d) combined with Theorem 5.5 imply that
Note that \(\alpha \) can be extended to a function in \(C_c([0,\infty ))\) which implies that the densities in the integrals in (8.1) and (8.2) are not singular. Hence, in the special cases considered here, we obtain a representation of functional intrinsic volumes as Hessian valuations with continuous densities and an additional term involving classical intrinsic volumes.
8.3 Retrieving the classical Steiner formula
As a further application of Theorem 1.4, we retrieve the classical Steiner formula (1.1) from (1.5). We need the following result, which shows how the classical intrinsic volumes can be retrieved from the functional intrinsic volumes.
Proposition 8.1
([16], Proposition 5.2) If \(\,0\le j \le n-1\) and \(\zeta \in D_{j}^{n}\), then
for every \(K\in {\mathcal {K}}^n\). If \(\zeta \in D_{n}^{n}\), then
for every \(K\in {\mathcal {K}}^n\).
Let \(r>0\) and choose u to be the convex indicator function of a convex body \(K\in {\mathcal {K}}^n\). We have
and therefore Theorem 1.4 combined with Lemma 3.2 and Proposition 8.1 implies that
for every \(K\in {\mathcal {K}}^n\) and \(\zeta \in D_{n}^{n}\), which gives the classical Steiner formula if \(\zeta (0)\ne 0\).
8.4 General functional Steiner formulas
We remark that the proof of Theorem 1.4 shows that Steiner formulas for convex functions are also obtained if we replace the convex indicator function \(\mathbf{I}_{B^n}\) by any radially symmetric, super-coercive, convex function. Similarly, the support function \(h_{B^n}\) in Theorem 2.3 can be replaced by any radially symmetric, finite-valued, convex function. However, in general such formulas do not give rise to all functional intrinsic volumes \({\text {V}}_{j,\zeta }^{*}\), that is, not all \(\zeta \in D_{j}^{n}\) will appear in the polynomial expansion. For example, if \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\cap C_+^2({\mathbb {R}}^n)\), then it easily follows from (2.1) and the definition of mixed discriminant that
for every \(\zeta \in D_{n}^{n}\) and \(r>0\). By continuity, (8.3) also holds for all \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). Here we use that \(D_{n}^{n}\subseteq D_{j}^{n}\) for every \(0\le j \le n\) to show that the functional intrinsic volumes appearing in (8.3) are well-defined. However, the classes \(D_{n}^{n}\) and \(D_{j}^{n}\) do not coincide if \(j<n\), which shows that not all functional intrinsic volumes \({\text {V}}_{n,\zeta _j}^{*}\) with \(\zeta _j\in D_{j}^{n}\) are obtained in this way.
This raises the question for which convex functions \(\phi :[0,\infty )\rightarrow {\mathbb {R}}\) we obtain all functional intrinsic volumes when we replace \(h_{B^n}\) by \(\phi \circ h_{B^n}\). Let \({\text {VConv}}_{j}({\mathbb {R}}^n; {\mathbb {R}})\) be the set of continuous, dually epi-translation and rotation invariant valuations on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) that are homogeneous of degree j. By Theorem 1.3, we know that
for \(0\le j\le n\). We obtain the following complete description if we use a regularity assumption for \(\phi \).
Theorem 8.2
Let \(\phi \in C^2([0,\infty ))\) be convex and such that \(\phi '(0)\ge 0\). For \(1\le j\le n-1\),
if and only if \(\phi '(0)>0\).
We require the following results for the proof of Theorem 8.2. The function \(v_t\) is defined in (3.6).
Lemma 8.3
Let \({{\,\mathrm{{\text {Z}}}\,}}_1,{{\,\mathrm{{\text {Z}}}\,}}_2:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) be continuous, dually epi-translation and rotation invariant valuations that are homogeneous of degree j with \(0\le j \le n\). If \({{\,\mathrm{{\text {Z}}}\,}}_1(v_t)={{\,\mathrm{{\text {Z}}}\,}}_2(v_t)\) for every \(t\ge 0\), then \({{\,\mathrm{{\text {Z}}}\,}}_1\equiv {{\,\mathrm{{\text {Z}}}\,}}_2\).
Proof
By Theorem 2.2, there exist \(\zeta _{1}, \zeta _{2}\in D_{j}^{n}\) such that
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\). If \(j=0\), then both \({{\,\mathrm{{\text {Z}}}\,}}_1\) and \({{\,\mathrm{{\text {Z}}}\,}}_2\) are constants, independent of v, and thus the statement is trivial. For \(1\le j\le n\), it follows from Lemma 3.4 and our assumptions on \({{\,\mathrm{{\text {Z}}}\,}}_1\) and \({{\,\mathrm{{\text {Z}}}\,}}_2\) that
for every \(t\ge 0\). By Lemma 3.2 this implies \(\zeta _1\equiv \zeta _2\) and thus \({{\,\mathrm{{\text {Z}}}\,}}_1\equiv {{\,\mathrm{{\text {Z}}}\,}}_2\). \(\square \)
We remark that it would be of great interest to find a proof of the previous lemma that does not require Theorem 2.2. In particular, this would provide a new strategy to prove the Hadwiger theorem for convex functions.
Lemma 8.4
Let \(1\le j\le n-1\), let \(\phi \in C^2([0,\infty ))\) be convex with \(\phi '(0)\ge 0\) and \(\beta \in C_c([0,\infty ))\). If the functional \(\,\bar{{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is given by
then
for \(t\ge 0\).
Proof
First, let \(\phi ,\psi \in C^2([0,\infty ))\) be convex and such that \(\psi '(0)=0\). We want to compute the mixed discriminant
For \(x\in {\mathbb {R}}^n\), set \(r:=\vert x\vert \). For \(r>0\), by the radial symmetry of \(\phi \circ h_{B^n}\) and \(\psi \circ h_{B^n}\) and by choosing a coordinate system such that \(e_n\) is parallel to x, we obtain
where \({\text {diag}}(\lambda _1,\dots ,\lambda _n)\) is the \(n\times n\) diagonal matrix with entries \(\lambda _1, \dots ,\lambda _n\) in the diagonal. Therefore, for \(\varepsilon >0\),
Using the previous expression and (3.1), we obtain, after some computations, that
Next, assume that \(\beta \in C^1_c([0,\infty ))\). By the previous step and Theorem 4.3 (c),
where we used polar coordinates, integration by parts and the condition \(\psi '(0)=0\). For \(t>0\), set
for \(r>0\). Note that for \(v_t\), we have
For \(t>0\), there exists a sequence of convex functions \(\psi _{t,j}\) converging to \(\psi _t\) uniformly in \([0,\infty )\) and such that \(\psi _{t,j}\in C^2([0,\infty ))\) and \(\psi _{t,j}'(0)=0\) for every j. Moreover, the sequence \(\psi _{t,j}\) can be chosen so that \(\psi '_{t,j}\) is uniformly bounded and converges pointwise to \(\psi _t'\) in \([0,\infty )\) except for \(r=t\). By (8.6), the weak continuity of Monge–Ampère measures, the fact that the support of \(\beta \) is bounded, and the dominated convergence theorem, we obtain that
Integration by parts gives
This equation, which has been proved in the case \(\beta \in C^1_c([0,\infty ))\), can now be extended to the case that \(\beta \in C_c([0,\infty ))\) by approximating \(\beta \) uniformly on its support by a sequence of functions in \(C^1_c([0,\infty ))\). \(\square \)
Theorem 8.2 follows from the next two propositions.
Proposition 8.5
Let \(\phi \in C^2([0,\infty ))\) be convex and let \(\phi '(0)>0\). If \(\,{{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) is a continuous, dually epi-translation and rotation invariant valuation that is homogeneous of degree j with \(1\le j\le n-1\), then there exists \(\beta \in C_c([0,\infty ))\) such that
for every \(v\in {\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\).
Proof
Given \(\alpha \in C_c([0,\infty ))\), define the function \(\beta :[0,\infty )\rightarrow {\mathbb {R}}\) as
where we use that \(\phi '(t)>0\) for every \(t\in [0,\infty )\). Also note that \(\beta \in C_c([0,\infty ))\). We claim that \(\beta \) is a solution of the equation
If we assume that \(\alpha \in C_c^1([0,\infty ))\), then also \(\beta \in C_c^1([0,\infty ))\) and (8.8) can be written in the form
Hence the claim is easily verified under the additional assumption on \(\alpha \). The general case is obtained by approximation.
For \(t\ge 0\), define
By Theorem 2.2, Lemmas 3.4 and 3.2, we know that \(\alpha \in C_c([0,\infty ))\). For this function \(\alpha \), define \(\beta :[0,\infty )\rightarrow {\mathbb {R}}\) by (8.7). Define \(\bar{{{\,\mathrm{{\text {Z}}}\,}}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) as
By Proposition 4.4, the functional \(\bar{{{\,\mathrm{{\text {Z}}}\,}}}\) is a continuous, dually epi-translation and rotation invariant valuation on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) that is homogeneous of degree j. By Lemma 8.4 and (8.8),
for every \(t\ge 0\). By Lemma 8.3, this implies that \({{\,\mathrm{{\text {Z}}}\,}}\equiv \bar{{{\,\mathrm{{\text {Z}}}\,}}}\). \(\square \)
Proposition 8.6
Let \(1\le j\le n-1\). If \(\phi \in C^2([0,\infty ))\) is convex and \(\phi '(0)=0\), then there exists a continuous, dually epi-translation and rotation invariant valuation \({{\,\mathrm{{\text {Z}}}\,}}:{\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\rightarrow {\mathbb {R}}\) that is homogeneous of degree j such that
is not verified by any \(\beta \in C_c([0,\infty ))\).
Proof
Let \(\alpha \in C^2_c([0,\infty ))\) be such that \(\alpha '(0)>0\). By Lemma 3.4 and Lemma 3.2, there exists a continuous, dually epi-translation and rotation invariant valuation \({{\,\mathrm{{\text {Z}}}\,}}\) on \({\mathrm{Conv}({\mathbb {R}}^n; {\mathbb {R}})}\) that is homogeneous of degree j such that
for \(t\ge 0\). Assume that there exists \(\beta \in C_c([0,\infty ))\) such that (8.4) is satisfied for this functional \({{\,\mathrm{{\text {Z}}}\,}}\). By Lemma 8.4, the function \(\beta \) has to verify (8.5). As \(\alpha \in C^2_c([0,\infty )\), we have \(\beta \in C^1_c([0,\infty ))\), and the equation takes the form
for \(t> 0\). Consequently,
for \(t>0\). By the conditions on \(\phi \) and \(\alpha \), we conclude that
which is a contradiction. \(\square \)
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Acknowledgements
M. Ludwig was supported, in part, by the Austrian Science Fund (FWF): P 34446 and F. Mussnig was supported by the Austrian Science Fund (FWF): J 4490-N.
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Colesanti, A., Ludwig, M. & Mussnig, F. The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge–Ampère measures. Calc. Var. 61, 181 (2022). https://doi.org/10.1007/s00526-022-02288-3
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DOI: https://doi.org/10.1007/s00526-022-02288-3