Abstract
In this paper we investigate the property of engulfing for Hconvex functions defined on the Heisenberg group \({\mathbb {H}^n}\). Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called \({\mathbb {H}^n}\)section, as well as a new condition of engulfing associated to the \({\mathbb {H}^n}\)sections, for an Hconvex function defined in \(\mathbb {H}^n.\) These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the \({\mathbb {H}^n}\)sections, with their engulfing property, will lead to the definition of a quasidistance in \({\mathbb {H}^n}\) in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round Hsections for an Hconvex function, and by its connection with the engulfing properties.
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1 Introduction
Given a convex function \(u:\mathbb {R}^n\rightarrow \mathbb {R}\), for every \(x_0\in \mathbb {R}^n,\) \(p\in \partial u(x_0),\) and \(s>0,\) we will denote by \(S_{u}(x_0,p,s)\) the section of u at \(x_0\) with height s, defined as follows
in case u is differentiable at \(x_0,\) we will denote the section by \(S_u(x_0,s),\) for short. The related notion of engulfing for convex functions, or, equivalently, for their sections, is essentially a geometric property, and it is based on a regular mutual behaviour of the sections of the function. We say that a convex function u satisfies the engulfing property (shortly, \(u \in E(\mathbb {R}^n,K)\)) if there exists \(K>1\) such that for any \(x\in \mathbb {R}^n,\) \(p\in \partial u(x),\) and \(s > 0\), if \(y\in S_u(x,p,s)\), then \(S_u(x,p,s)\subset S_u(y,q,Ks),\) for every \(q\in \partial u(y)\).
The functions u in the class \(E(\mathbb {R}^n,K)\) have been studied in connection with the solution to the MongeAmpère equation \(\mathrm {det} \,D^2u=\mu ,\) where \(\mu \) is a Borel measure on \(\mathbb {R}^n.\) In this framework, a \(\mathcal {C}^{1,\,\beta }\)estimate for the strictly convex, generalized solutions to the MongeAmpère equation was proved by Caffarelli ([7, 8]), under the assumption that the measure \(\mu \) satisfies a suitable doubling property (see the exhaustive book by Gutiérrez [19]). This doubling property is actually equivalent to the geometric property of engulfing for the solution u.
Another issue is related to the properties enjoyed by the family of sections \(\{S_u(x,s)\}_{\{x\in \mathbb {R}^n,\, s>0\}},\) in case u is a convex differentiable function in \(E(\mathbb {R}^n, K).\) In [1], it is shown that, in this case, one can define a quasidistance d on \(\mathbb {R}^n\) as follows:
In addition, if \(B_d(x,r)\) is a dball of center x and radius r, then
In the archetypal case \(u(x)=\Vert x\Vert ^2,\) with \(x\in \mathbb {R}^n\), one has \(S_u(x,s)=B^{\mathbb {R}^n}(x,\sqrt{s}),\) and hence the family of sections of u consists of the usual balls in \(\mathbb {R}^n\).
In the case of convex functions defined in a Carnot group \(\mathbf{G} \), in [13] Capogna and Maldonado introduced some appropriate geometric objects, that can be considered as the subRiemmannian analogue of the classical sections, as well as a naturally related notion of horizontal engulfing. Given a horizontally convex function \(\varphi : \mathbf{G} \rightarrow \mathbb {R},\) \(\xi _0\in \mathbf{G} ,\) \(p\in \mathbb {R}^{m_1},\) \(s>0,\) the section \(S_u^H(\xi _0,p,s)\) (Hsections, from now on, where H stands for horizontal) is defined as follows:
where \(V_1\cong \mathbb {R}^{m_1}\) is the first layer of the stratification of the Lie algebra of \(\mathbf{G} \); in case \(\varphi \) is horizontally differentiable at \(\xi _0,\) we will denote such Hsection by \(S_\varphi ^H(\xi _0,s),\) for short. The mentioned authors say that a horizontally convex and differentiable function \(\varphi \) satisfies the engulfing property if there exists \(K>1\) such that, for every \(\xi ,\, \xi '\in \mathbf{G} \) and \(s>0\), if \(\xi '\in S_\varphi ^H(\xi ,s)\), then \(\xi \in S_\varphi ^H(\xi ',Ks)\). Let us stress that the definition of Hsection in (1.4) and the notion of engulfing are affected by the subRiemannian structure exactly as the notion of horizontal convexity; more precisely, they rely upon the behaviour of the function on the horizontal lines and planes. In [13] it is proved that a strictly convex and everywhere differentiable function on a Carnot group, satisfying this horizontal version of engulfing, belong to the Folland–Stein class \(\Gamma ^{1+1/K},\) i.e., the horizontal derivatives \(X_i\varphi \) are 1/KHölder continuous with respect to any leftinvariant and homogeneous pseudonorm in the group. The key point in their argument is a reduction of the general discussion to the onedimensional case. As a matter of fact, the “dimension” of the Hsections in (1.4) is the dimension of the first layer of the stratification of the Lie algebra of the group; in particular, these Hsections have empty interior. This fact prevents from building a quasidistance as in (1.2) starting from the family of sections associated to every point of the group.
In this paper we focus on horizontally convex functions \(\varphi \) (Hconvex functions) on the Heisenberg group \(\mathbb {H}^n,\) that is the simplest Carnot group of step 2. Our main purpose is to overcome the dimensional gap between the Hsections, defined in [13], and the balls related to any quasidistance in \(\mathbb {H}^n,\) by introducing and studying a different notion of section. Our idea takes inspiration from the notion of Hsection in (1.4), together with the property that any pair of points in \(\mathbb {H}^n\) can be joined by at most three consecutive horizontal segments. These facts lead us to define fulldimensional sections that arise as a sort of composition in three steps of “thin” Hsections. These new objects will be called \({\mathbb {H}^n}\)section, and will be denoted by \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi _0,p,s)\) (for the precise definition of \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi _0,p,s),\) see Definition 5.1). For these \({\mathbb {H}^n}\)sections, we introduce the following engulfing condition:
Definition 1.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. We say that \(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n},K)\) if there exists \(K > 1\) such that for any \(\xi \in {\mathbb {H}^n},\ p\in \partial _H \varphi (\xi )\) and \(s > 0\), if \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,p,s)\), then
for every \(q\in \partial _H \varphi (\xi ')\).
It is obvious that a function which satisfies this engulfing property \(E({\mathbb {H}^n},K)\), satisfies the engulfing property introduced by Capogna and Maldonado.
The study of this new notion of engulfing for \({\mathbb {H}^n}\)sections of full dimension requires a mix of tools and properties inherited by the Euclidean case \(\mathbb {R}^n\), both for the simplest case \(n=1,\) and for the knotty case \(n>1\). Following the idea in [21] and, in particular, the equivalence between iii. and iv in Theorem 7.1 below, we introduce and study a horizontal notion of round sections for the Hsections (see Definition 3.1). We prove that every Hconvex function with round Hsections satisfies the engulfing property \(E({\mathbb {H}^n},K)\) in Definition 1.1.
Let us summarize our results as follows:
Theorem 1.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections, then

i.
\(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n},K)\); consequently, in the class of Hconvex functions with round Hsections, the engulfing for Hsections and the engulfing for \({\mathbb {H}^n}\)sections are equivalent properties;

ii.
the function \(d_\varphi :{\mathbb {H}^n}\times {\mathbb {H}^n}\rightarrow [0,+\infty )\) defined by
$$\begin{aligned} d_\varphi (\xi ,\xi ')=\inf \left\{ s>0:\ \xi \in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',s),\ \xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\right\} \end{aligned}$$is a quasidistance in \({\mathbb {H}^n}\); moreover, for the \(d_\varphi \)balls, an \({\mathbb {H}^n}\)version of the inclusions in (1.3) holds true (see (6.10) below).
Here, the archetypal example in \(\mathbb {H}\) of the Hconvex function \(\varphi (x,y,t)=x^2+y^2\) gives \(S_\varphi ^\mathbb {H}(\xi ,s)=\widetilde{B}(\xi ,\sqrt{s}),\) that is, the family of \(\mathbb {H}\)sections of \(\varphi \) consists of the \(\widetilde{B}\)balls of a leftinvariant and homogeneous distance \(\widetilde{d}\) (see (5.2) and Example 6.1).
The property of round Hsections is actually stronger than the horizontal engulfing; we are able to provide an example of an Hconvex function which satisfies the horizontal engulfing property but does not have round Hsections, and this phenomenon appears also in the Euclidean case, if \(n>1\). Nevertheless, the main issue of the result above relies upon the dimensional gap between the assumptions, where a purely horizontal property is required, and the final result, where fulldimensional sets are involved.
The paper is organized as follows. In Sect. 2 we recall some results related to the engulfing property for a function defined in \(\mathbb {R}^n,\) together with the structure of \(\mathbb {H}^n\) and the notion of horizontal convexity. In Sect. 3 we introduce the Hsections, and we show that round Hsections and controlled Hslope are equivalent property for these Hsections (see Theorem 3.1). In Sect. 4 we characterize the functions with the engulfing property E(H, K), and prove that the two properties introduced in the previous section are sufficient conditions for a function to be in E(H, K). In Sect. 5 we move to the notion of \({\mathbb {H}^n}\)sections and the related engulfing property as in Definition 1.1, and we prove Theorem 1.1 i. In Sect. 6 we prove Theorem 1.1 ii. and provide a concrete example. In the final section we list some open questions.
2 Preliminary Notions and Results
In the paper, we will deal with Hconvex functions defined on the Heisenberg group \(\mathbb {H}^n.\) As we will see later, the notion of Hconvexity requires that, for every point \(\xi \in \mathbb {H}^n,\) one looks at the behaviour of the function under two points of view. The first one is onedimensional, since the restriction of the function to any horizontal line \(\{\xi \circ \exp tv\}_{t\in \mathbb {R}},\) with \(v\in V_1,\) is an ordinary convex function; the second one is 2ndimensional, according to the fact that \(v\in V_1\cong \mathbb {R}^{2n},\) or, equivalently, the horizontal lines through \(\xi \) span the 2ndimensional horizontal plane \(H_\xi .\) For these reasons, the first part of this section will be devoted to some results related to the engulfing property of convex functions \(u:\mathbb {R}^n\rightarrow \mathbb {R}\), both in the case \(n=1,\) and in the case \(n\ge 2.\) In the second part we will recall the notion of Hconvexity, together with some related results, for functions defined on the Heisenberg group \({\mathbb {H}^n}\).
2.1 The Engulfing Property for Convex Functions in \({\pmb {\mathbb {R}}}^n\)
Let us concentrate, first, on the onedimensional case, i.e. \(n=1\). The following characterization holds (see Theorem 2 in [18], Theorem 5.1 in [14]):
Theorem 2.1
Let \(u:\mathbb {R}\rightarrow \mathbb {R}\) be a strictly convex and differentiable function. The following are equivalent:

i.
\(u\in E(\mathbb {R},K)\), for some \(K>1\);

ii.
there exists a constant \(K'>1\) such that, if \(x,y\in \mathbb {R}\) and \(s>0\) verify \(x\in S_u(y,s),\) then \(y\in S_u(x,K's);\)

iii.
there exists a constant \(K''> 1\) such that, for any \(x,\ y\in \mathbb {R},\)
$$\begin{aligned}&\displaystyle \frac{K''+1}{K''}\left( u(y)u(x)u'(x)(yx)\right) \le (u'(x)u' (y))(xy)\nonumber \\&\quad \le (K''+1)\left( u(y)u(x)u'(x)(yx)\right) . \end{aligned}$$(2.1)
As a matter of fact, the assumption of differentiability in the theorem above can be removed, as proved in [11]:
Theorem 2.2
Let \(u:\mathbb {R}\rightarrow \mathbb {R}\) be a convex function, with bounded sections, satisfying the engulfing property. Then, u is strictly convex and is in \({\mathcal C}^1(\mathbb {R})\).
Given a strictly convex differentiable function \(u:\mathbb {R}\rightarrow \mathbb {R},\) one can consider the associated MongeAmpère measure \(\mu _{u}\) defined on any Borel set \(A\subset \mathbb {R}\) by
where \(\cdot \) denotes the Lebesgue measure. We say that the measure \(\mu _{u}\) has the (DC)doubling property if there exist constants \(\alpha \in (0,1)\) and \(C>1\) such that
for every section \(S_{u}(x,s)\) (here \(\alpha S_{u}(x,s)\) is the open convex set obtained by \(\alpha \)contraction of \(S_{u}(x,s)\) with respect to its center of mass). In [20] and [17] it was shown that the (DC)doubling property of the measure \(\mu _{u}\) is equivalent to the engulfing property for the function u; in particular, given u in \(E(\mathbb {R},K),\) the constants \(\alpha \) and C in (2.2) depend only on K. A Radon measure \(\mu \) is doubling if and only if there exists a constant A such that
for any congruent cubes \(Q_1\) and \(Q_2\) with nonempty intersection (see, for example, [22]). We recall that two subsets of \(\mathbb {R}\) are called congruent if there exists an isometry of \(\mathbb {R}\) that maps one of them onto the other. Since every open and bounded interval in \(\mathbb {R}\) is a particular section for u, the (DC)doubling property of \(\mu _{u}\) is trivially equivalent to the fact that \(\mu _{u}\) is a doubling measure. In particular, the constant A depends only on K. Now, noticing that \(\mu _{u}((x,x+r))=u'(x+r)u'(x)\), by (2.1) we obtain
These arguments show the central role of the function \((x,r)\mapsto u(x+r)u(x)u'(x)r\) in our paper. More precisely in [14] (see Theorem 5.5) the authors prove the following:
Theorem 2.3
Let \(u:\mathbb {R}\rightarrow \mathbb {R}\) be a strictly convex and differentiable function. Then \(u\in E(\mathbb {R},K)\) if and only if there exist two constants \(A_1>1\) and \(A_2>1,\) both of them depending on \(K_,\) such that
Condition (2.4) says that u is essentially symmetric around every point, and condition (2.5) says that it satisfies the socalled \(\Delta _2\) condition at each point in \(\mathbb {R}\).
Hence, the behaviour of the measure \(\mu _{u}\) is related to the functions \(m_u,\ M_u:\mathbb {R}\times \mathbb {R}^+\rightarrow \mathbb {R}^+\) defined by
for every \(x\in \mathbb {R},\ r\in \mathbb {R}^+\). These functions will be naturally extended to the ndimensional case and in \({\mathbb {H}^n}\), and will play a crucial role in the investigation of the engulfing for Hconvex functions.
For every fixed \(x\in \mathbb {R},\) denote by \(u_x\) the function
Then, \(M_u(x,r)\in \{u_x(\pm r)\},\) and \(M_u(x,2r)\in \{u_x(\pm 2r)\}.\) Let us suppose, for instance, that the following equalities hold true:
Then, by (2.4) and (2.5), we obtain
The other possible combinations can be treated similarly, and we obtain the following fundamental estimates:
Remark 2.1
Let \(u\in E(\mathbb {R},K)\) be a strictly convex and differentiable function. Then,
where \(B_1,\ B_2\) and \(B_3\) depend only on K (and \(B_i>1\)).
It is worthwhile to note that inequality (2.10) is false if \(n\ge 2,\) despite the engulfing property holds; the function in (4.9), due to Wang, will provide a counterexample to this phenomenon.
The next result provides another estimate for the function \(m_u\):
Proposition 2.1
Let \(u\in E(\mathbb {R},K)\) be a convex function with bounded sections. Then,
with \(B_4>1\) which depends only on K.
Proof
Let us fix \(x\in \mathbb {R}\). The function \(u_x\) defined in (2.7) is strictly convex and differentiable (see [11]), and belongs to \(E(\mathbb {R},K)\); moreover,
where \(K''\) depends only on K (for all the details, see Theorem 4 and its proof in [18]). Hence, for every fixed \(r>0\), the Gronwall inequality gives
Therefore, we obtain that \(u_x(\pm 2r)\ge 2^{\frac{K''+1}{K''}}u_x(\pm r).\) Let \(m_u(x,r)=u_x(r).\) Then, \(B_4 m_u(x,r)\le u_x(2r).\) Suppose that \(B_4m_u(x,r)>u_x(r).\) In this case, \(u_x(2r)\ge B_4 u_x(r),\) and thus \(B_4u_x(r)<B_4 m_u(x,r),\) a contradiction. Then, (2.11) follows. \(\square \)
Let us now move to the case \(n\ge 2.\) Given a differentiable function \(u:\mathbb {R}^n\rightarrow \mathbb {R},\) as in the onedimensional case (2.6), the functions \(m_u,\ M_u:\mathbb {R}^n\times \mathbb {R}^+\rightarrow \mathbb {R}^+\) are defined by
for every \(x\in \mathbb {R}^n,\ r\in \mathbb {R}^+\).
Let us recall the following property, that will be critical when dealing with the engulfing in \(\mathbb {H}^n.\)
Definition 2.1
(see Definition 2.1 in [21]) Let \(u:\mathbb {R}^n\rightarrow \mathbb {R}\) be a convex function. We say that u has round sections if there exists a constant \(\tau \in (0,1)\) with the following property: for every \(x\in \mathbb {R}^n,\) \(p\in \partial u(x),\) and \(s>0,\) there is \(R>0\) such that
In [21] (see Theorem 7.1 below) it is proved that a convex function \(u:\mathbb {R}^n\rightarrow \mathbb {R}\) has round sections if and only if u is differentiable, but not affine, and has controlled slope, i.e., there exists a constant \(H\ge 1\) such that
This equivalence is quantitative, in the sense that the constants involved in each statement depend only on each other and n, but not on u. Furthermore, if \(u:\mathbb {R}^n\rightarrow \mathbb {R}\) satisfies one of the two equivalent conditions above, then \(u\in E(\mathbb {R}^n,K),\) for a suitable \(K>1\) (see Theorem 3.9 in [21]). Let us finally notice that condition (2.12) is the ndimensional version of condition (2.10): in the case \(n\ge 2\), hence, the controlled slope for a function, or, equivalently, the property of round sections, is only a sufficient condition for a function to have the engulfing property.
2.2 Convexity in the Heisenberg Group \({\pmb {\mathbb {H}}^\mathbf{n} }\)
The Heisenberg group \({\mathbb {H}^n}\) is the simplest Carnot group of step 2. We will recall some of the notions and background results used in the sequel. We will focus only on those geometric aspects that are relevant to our paper. For a general overview on the subject, we refer to [6] and [12].
The Lie algebra \(\mathfrak {h}\) of \(\mathbb H^n\) admits a stratification \( \mathfrak {h}=V_1\oplus V_2\) with \(V_1=\text { span}\{X_{i},\, Y_i;\ 1\le i\le n\}\) being the first layer of the socalled horizontal vector fields, and \(V_2=\text { span}\{T\}\) being the second layer which is onedimensional. We assume \([X_i,Y_i]=4T\) and the remaining commutators of basis vectors vanish. The exponential map \(\exp :\mathfrak {h}\rightarrow \mathbb H^n\) is defined in the usual way. By these commutator rules we obtain, using the BakerCampbellHausdorff formula, that \({\mathbb {H}^n}\) can be identified with \(\mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R}\) endowed with the noncommutative group law given by
where \(x,y,x'\) and \(y'\) are in \(\mathbb {R}^n\), t and \(t'\) in \(\mathbb {R}\), and where \('\cdot '\) is the inner product in \(\mathbb {R}^n\). Let us denote by e the neutral element in \({\mathbb {H}^n}.\) Transporting the basis vectors of \(V_1\) from the origin to an arbitrary point of the group by a lefttranslation, we obtain a system of leftinvariant vector fields written as first order differential operators as follows
Via the exponential map \(\exp :\mathfrak {h}\rightarrow \mathbb {H}\) we identify the vector \(\sum _{i=1}^n(\alpha _i X_i+\beta _i Y_i)+\gamma T\) in \(\mathfrak {h}\) with the point \((\alpha _1,\ldots ,\alpha _n, \beta _1,\ldots ,\beta _n, \gamma )\) in \({\mathbb {H}^n};\) the inverse \(\xi : {\mathbb {H}^n}\rightarrow \mathfrak {h}\) of the exponential map has the unique decomposition \(\xi =(\xi _1,\xi _2),\) with \(\xi _i:{\mathbb {H}^n}\rightarrow V_i,\) and we identify \(V_1\) with \(\mathbb {R}^{2n}\) when needed.
For every positive \(\lambda ,\) the nonisotropic Heisenberg dilation \(\delta _{\lambda }: \mathbb {H}^{n}\rightarrow \mathbb {H}^{n}\) is defined by \(\delta _{\lambda }(x,y,t) = (\lambda x,\lambda y, \lambda ^{2}t)\). Let \(N(x,y,t)=((\Vert x\Vert ^2+\Vert y\Vert ^2)^2+t^2)^\frac{1}{4}\) be the gauge norm in \(\mathbb H^n\). The function \(d_g:\mathbb {H}^n\times \mathbb {H}^n\rightarrow [0,+\infty )\) defined by
satisfies the triangle inequality, thereby defining a distance on \({\mathbb {H}^n}\): this distance is the socalled KorányiCygan distance which is leftinvariant and homogeneous, i.e. \(d_g(\delta _\lambda (\xi ),\delta _\lambda (\xi '))=\lambda d_g(\xi ,\xi ')\) for every \(\lambda >0,\) \(\xi ,\xi '\in {\mathbb {H}^n}\). We will set \(d_g(e,\xi )=\Vert \xi \Vert _g\) for every \(\xi \in \mathbb {H}^n\). The KorányiCygan ball of center \(\xi _0\in \mathbb H^n\) and radius \(r>0\) is given by \(B_{g}(\xi _0,r)=\{\xi \in {\mathbb {H}^n}:\ d_g(\xi _0,\xi )\le r\}.\)
The horizontal structure relies on the notion of horizontal plane. Given \(\xi _0\in {\mathbb {H}^n}\), the horizontal plane \(H_{\xi _0}\) associated to \(\xi _0=(x_0,y_0,t_0)\) is the plane in \({\mathbb {H}^n}\) defined by
This is the plane spanned by the horizontal vector fields \(\{X_i,\ Y_i\}_i\) at the point \(\xi _0\); note that \(\xi '\in H_{\xi }\) if and only if \(\xi \in H_{\xi '}\). A horizontal segment is a convex subset of a horizontal line, which is a line lying on a horizontal plane \(H_\xi \) and passing though the point \(\xi \in {\mathbb {H}^n}\); if \(\xi '\in H_{\xi }\), with \(\xi '\not =\xi \), then \(H_\xi \cap H_{\xi '}\) is a horizontal line.
Let \(\Omega \subset \mathbb H^n\) be an open set. The main idea of the analysis in the Heisenberg group is that the regularity properties of functions defined in \(\mathbb {H}^n\) can be expressed in terms only of the horizontal vector fields (2.13). In particular, the appropriate notion of gradient for a function is the socalled horizontal gradient, which is defined as the 2nvector \( \nabla _H\varphi (\xi )= \left( X_1\varphi (\xi ),...,X_n\varphi (\xi ), Y_1\varphi (\xi ),...,Y_n\varphi (\xi )\right) \) for a function \(\varphi \in \Gamma ^1(\Omega )\). Here, \(\Gamma ^k(\Omega )\) denotes the Folland–Stein space of functions having continuous derivatives up to order k with respect to the vector fields \(X_i\) and \(Y_i,\) \(i\in \{1,...,n\}\). We say that \(\varphi : \Omega \rightarrow \mathbb {R}\) is Hdifferentiable at \(\xi \), if there exists a mapping \(D_H \varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) which is Hlinear, i.e. \(D_H \varphi (x,y,t)=D_H \varphi (x,y,0)\) for every \((x,y,t)\in {\mathbb {H}^n}\), such that \(\varphi (\xi \circ \xi ') = \varphi (\xi )+ D_H \varphi (\xi ')+o(\Vert \xi '\Vert _g)\); the vector associated to \(D_H \varphi \) with respect to the fixed scalar product is the horizontal gradient \( \nabla _H \varphi (\xi )\).
For general nonsmooth functions \(\varphi :\Omega \rightarrow \mathbb R,\) the horizontal subdifferential \(\partial _H \varphi (\xi _0)\) of \(\varphi \) at \(\xi _0\in \Omega \) is given by
where \({\text {Pr}_1}:\mathbb {H}^n\rightarrow \mathbb {R}^{2n}\) is the projection defined by \({\text {Pr}_1}(\xi )={\text {Pr}_1}(x,y,t)=(x,y)\). It is easy to see that if \(\varphi \in \Gamma ^1(\Omega )\) and \(\partial _H\varphi (\xi )\ne \emptyset \), then \(\partial _H\varphi (\xi )=\{\nabla _H \varphi (\xi )\}\). A function \(\varphi :\Omega \rightarrow \mathbb R\) is called \(H\)subdifferentiable on \(\Omega \) if \(\partial _H \varphi (\xi )\ne \emptyset \) for every \(\xi \in \Omega .\)
A central object of study within this paper is provided by the Hconvex functions. First of all, we recall that a set \(\Omega \subset \mathbb H^n\) is said to be horizontally convex (Hconvex) if, for every \(\xi _1,\xi _2\in \Omega ,\) with \(\xi _1\in H_{\xi _2}\) and \(\lambda \in [0,1]\), we have \(\xi _1\circ \delta _\lambda (\xi _1^{1}\circ \xi _2)\in \Omega \). It is clear that if \(\Omega \) is convex (i.e. it is convex in the \(\mathbb {R}^{2n+1}\)sense), then it is also Hconvex. Given a function \(\varphi :\Omega \rightarrow \mathbb R,\) where \(\Omega \) is Hconvex, there are several equivalent ways to define the concept of Hconvexity for \(\varphi .\) The most intuitive one is to require the classical convexity of the function when restricted to any horizontal line within \(\Omega .\) The same definition can be rephrased by considering the group operation: the function \(\varphi :\Omega \rightarrow \mathbb R\) is said to be Hconvex if, for every \(\xi _1,\xi _2\in \Omega \) with \(\xi _1\in H_{\xi _2}\) and \(\lambda \in [0,1]\), we have that
If the strict inequality holds in (2.14), for every \(\xi _1\ne \xi _2\) and \(\lambda \in (0,1)\), then \(\varphi \) is said to be strictly Hconvex. Hconvex functions have been extensively studied in the last few years; their characterizations, as well as their regularity properties, like their continuity, for instance, will come into play through the paper, and we refer to [5, 9, 15, 23]. Let us recall, in particular, that \(\varphi :\mathbb {H}^n\rightarrow \mathbb {R}\) is Hconvex if and only if \(\varphi \) is Hsubdifferentiable.
3 HConvex Functions with Round HSections and with Controlled HSlope
As already seen in the Introduction, a horizontal notion of section was given in [13] for functions defined on a general Carnot group \(\mathbf{G} .\) We will consider the particular case \(\mathbf{G} =\mathbb {H}^n.\)
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function, and let us fix \(\xi _0\in {\mathbb {H}^n},\) \(p_0\in \partial _H \varphi (\xi _0),\) and \(s>0.\) The Hsection of \(\varphi \) at \(\xi _0,\) \(p_0,\) with height s, is the set
If \(\varphi \) is Hdifferentiable, then \(\partial _H \varphi (\xi _0)=\{\nabla _H \varphi (\xi _0) \},\) and we simply write \(S_\varphi ^H(\xi _0,s)\) for the corresponding Hsection. For every fixed \((\xi _0,p_0,s),\) the set \(S_\varphi ^H(\xi _0,p_0,s)\) is Hconvex, and is contained in a horizontal plane; this dimensional gap between Hsections and open sets in \({\mathbb {H}^n}\) is a crucial difference with respect to the Euclidean case.
In this section we essentially introduce the notions of round Hsections (see Definition 3.1) and controlled Hslope (see Definition 3.2), proving their equivalence (see Theorem 3.1). Let us emphasize that these two properties for an Hconvex function are horizontal properties, i.e. they give information on the behaviour of the function only when restricted to the horizontal planes, exactly as the notion of Hsection, Hconvexity and Hsubdifferential.
In the following of the paper, for every function \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R},\) and for every \(\xi _0\in {\mathbb {H}^n}\), \(p_0\in \partial _H\varphi (\xi _0)\) and \(v_0\in V_1\setminus \{0\}\), we will consider the functions \(\varphi _{\xi _0,p_0}:{\mathbb {H}^n}\rightarrow \mathbb {R}\) and \(\widehat{\varphi }_{\xi _0,v_0}:\mathbb {R}\rightarrow \mathbb {R}\) defined by
If \(\varphi \) is Hdifferentiable, then we will set \(\varphi _{\xi _0,\nabla _H\varphi (\xi _0)}=\varphi _{\xi _0}\). The following result holds:
Proposition 3.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be a strictly Hconvex function. Then, all its Hsections are bounded sets.
Proof
For every \(\xi _0\in {\mathbb {H}^n}\) and \(v\in V_1\setminus \{0\}\) let us consider the function \(\widehat{\varphi }_{\xi _0,v}\) as in (3.3). By contradiction, let us suppose that there exists a sequence \(\{(v_n,\alpha _n)\}_n\), with \(v_n\in V_1,\ \Vert v_n\Vert =1,\ \alpha _n\rightarrow +\infty ,\) such that \(\xi _0\circ \exp (\alpha _n v_n)\in S^H_\varphi (\xi _0,p_0,s)\). Clearly, there exists a subsequence such that \(v_n\rightarrow v_0\in V_1.\)
Let us denote by \(\alpha _0=\sup \left\{ \alpha \ge 0:\ \xi _0\circ \exp (\alpha v_0)\in \overline{S^H_\varphi (\xi _0,p_0,s)}\right\} .\) If \(\alpha _0=+\infty \), then the section \(S_{\widehat{\varphi }_{\xi _0,v}}(0,s)\) of the function \(\widehat{\varphi }_{\xi _0,v}\) is unbounded; this is impossible, since \(\widehat{\varphi }_{\xi _0,v}\) is strictly convex. Let \(s_0\) be finite, and let us consider the function \(\varphi _{\xi _0,p_0}\) in (3.2); the set \(A=\{\xi \in {\mathbb {H}^n}:\ \varphi _{\xi _0,p_0}(\xi )\le s\}\) is Hconvex, since the function \(\varphi _{\xi _0,p_0}\) is Hconvex. Now, the previous arguments give
This contradicts Theorem 1.4 in [3]. \(\square \)
The next definition is related to a purely geometric property of the sections, and it will play a crucial role in the following of the paper.
Definition 3.1
We say that an Hconvex function \(\varphi : {\mathbb {H}^n}\rightarrow \mathbb {R}\) has round Hsections if there exists a constant \(K_0\in (0, 1)\) with the following property: for every \(\xi \in {\mathbb {H}^n}\), \(p\in \partial _H \varphi (\xi )\) and \(s>0,\) there exists \(R>0\) such that
In particular, (3.4) implies that every Hsection of a function with round Hsections is a bounded set. Clearly, Definition 3.1 is the \({\mathbb {H}^n}\)version of Definition 2.1; let us stress that it relies upon the subriemannian structure of \({\mathbb {H}^n}\) since, for every point \(\xi ,\) we restrict our attention only to the horizontal plane \(H_\xi \).
Remark 3.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be Hconvex, and consider the convex function \(\widehat{\varphi }_{\xi _0,v}:\mathbb {R}\rightarrow \mathbb {R}\) defined by (3.3). Then, if the nonempty convex set \(\partial _H\varphi (\xi _0)\) is not a singleton, there exists \(v\in V_1\) such that \(\partial \widehat{\varphi }_{\xi _0, v}(0)\) is not a singleton. Indeed, suppose that \(p+\lambda q\in \partial _H\varphi (\xi _0),\) for every \(\lambda \in [0,1],\) with \(q\ne 0.\) Then, by taking \(v=q,\) we have that
Hence \(p\cdot q+\lambda \Vert q\Vert ^2\in \partial \widehat{\varphi }_{\xi _0,q}(0)\) for every \(\lambda \in [0,1].\) This implies that, if \(\widehat{\varphi }_{\xi _0,v}\) is differentiable at 0 for every \(v\in V_1,\) then \(\varphi \) is Hdifferentiable at \(\xi _0.\)
In the previous remark and in the following result, the Hconvexity plays a fundamental role in order to obtain some regularity properties of the function involved.
Proposition 3.2
If \(\varphi : {\mathbb {H}^n}\rightarrow \mathbb {R}\) is an Hconvex function with round Hsections, then it is Hdifferentiable and strictly Hconvex. Moreover, there exists a constant C such that, for every \(\xi _0\in {\mathbb {H}^n}\) and \(v\in V_1,\) we have
where the constant C depends only on \(K_0\) in (3.4).
Proof
First of all note that, for every \(\xi _0\in {\mathbb {H}^n}\) and \(v\in V_1\setminus \{0\},\) the function \(\widehat{\varphi }_{\xi _0,v}\) defined in (3.3) is convex, with round sections (with constant \(K_0\)). Therefore, Lemma 3.2 in [21] implies that it is differentiable and strictly convex. In particular, \(\varphi \) is strictly Hconvex. Let us first show that \(\varphi \) is Hdifferentiable at \(\xi _0\in {\mathbb {H}^n}\). Since \(\varphi \) is Hconvex, this is equivalent to prove that the nonempty convex set \(\partial _H\varphi (\xi _0)\) is a singleton (see Theorem 4.4, Prop. 5.1 in [9], Theorem 1.4 in [23]). Suppose, by contradiction, that \(\partial _H \varphi (\xi _0)\) is not a singleton; then, by Remark 3.1, there exists \(v\in V_1\) such that \(\partial \widehat{\varphi }_{\xi _0,v}(0)\) is not a singleton. This contradicts the fact that \(\widehat{\varphi }_{\xi _0,v}(0)\) is differentiable.
Finally, taking into account that the function \(\widehat{\varphi }_{\xi _0,v}\) is convex, differentiable and with round sections with constant \(K_0\) , for every \(\xi _0\in {\mathbb {H}^n}\) and \(v\in V_1,\) again, by Lemma 3.2 in [21], one has that there exists a constant C depending only on \(K_0\) such that
\(\square \)
In the sequel, given an Hdifferentiable function \(\varphi : {\mathbb {H}^n}\rightarrow \mathbb {R}\), we will deal with the functions \(m_\varphi ^H,\ M_\varphi ^H:{\mathbb {H}^n}\times \mathbb {R}^+\rightarrow \mathbb {R}^+\) that will take the place in \(\mathbb {H}^n\) of the functions \(m_u\) and \(M_u\) in \(\mathbb {R}^n.\) They are defined as follows:
for every \(\xi \in {\mathbb {H}^n},\ r>0\).
A simple exercise shows that, if \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) is an Hdifferentiable and strictly Hconvex function, then for every \( \xi \in {\mathbb {H}^n},\) and \(r>0\),
The next definition, inherited from the corresponding one in \(\mathbb {R}^n\) (see (2.12)), pertains to the mutual behaviour of \(m_\varphi ^H\) and \(M_\varphi ^H,\) always from a horizontal point of view:
Definition 3.2
We say that an Hconvex function \(\varphi : {\mathbb {H}^n}\rightarrow \mathbb {R}\) has controlled Hslope if \(\varphi \) is Hdifferentiable, and there exists a constant \(K_1>0\) such that, for every \(\xi \in {\mathbb {H}^n}\) and \(r>0,\)
Like in the Euclidean case (see Theorem 7.1) controlled Hslope and round Hsections properties are strictly related:
Theorem 3.1
Let \(\varphi : {\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. The following conditions are equivalent:

a.
\(\varphi \) is an Hdifferentiable function, with bounded Hsections and controlled Hslope;

b.
\(\varphi \) has round Hsections.
Moreover, the constants \(K_0\) and \(K_1\) in (3.4) and in (3.7) are related, and they depend only on \(\varphi \).
Proof
Let a. be true. Let \(S^H_\varphi (\xi _0, s)\) be a bounded Hsection, and let \(R=\max \left\{ d_g(\xi ,\xi _0):\ \xi \in \overline{S^H_\varphi (\xi _0, s)}\right\} \). Pick a point \(\xi '\) such that \(d_g(\xi ',\xi _0)=R;\) then, \(\xi '\in \partial {S^H_\varphi (\xi _0, s)}\) and \(\xi '=\xi _0\circ \exp v'\). From the Hconvexity of \(\varphi _{\xi _0}\) on \(\mathbb {H}^n,\) we have that
where \(K_1\) is as in (3.7). Now, for every \(\xi \in H_{\xi _0}\) such that \(d_g(\xi ,\xi _0)=\frac{R}{K_1},\) by (3.7) we have
Hence,
Suppose now that condition b. holds true. Proposition 3.2 entails that \(\varphi \) is Hdifferentiable. Consider \(K_0\) as in (3.4), and fix \(\xi \in {\mathbb {H}^n}\) and \(r>0\): we have to prove (3.7), where \(K_1\) is uniform, i.e. it does not depend on \(\xi \) and r. Set \(s=m^H_\varphi (\xi ,r)\) and define
Since \(\varphi \) has round Hsections, \({\mathcal R}\) is not empty. Set \(R=\min {\mathcal R}\); trivially, \(R=r,\) and
for every \(v\in V_1,\ \Vert v\Vert =R.\) The two relations above imply that
Take \(\alpha \in \mathbb {N}\) such that \(K_0>2^{\alpha },\) and note that relation (3.5) implies
for every \(R_1>0,\) where C depends only on \(K_0\). By iterating this relation, we obtain
This last inequality, together with (3.8), leads to the assertion, with \(K_1=C^{\alpha }\) in (3.7). \(\square \)
In the next result we investigate the properties of the function \({m}_\varphi ^H,\) in order to shed some light on a finer behaviour of the Hsections.
Proposition 3.3
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hdifferentiable and strictly Hconvex function. For every fixed \(\xi \in {\mathbb {H}^n}\), the function \(r\mapsto m_\varphi ^H(\xi ,r)\) is strictly increasing, continuous, and it goes to \(+\infty ,\) if \(r\rightarrow +\infty \). Then, the function \( m_\varphi ^H(\xi ,\cdot ):[0,+\infty )\rightarrow [0,+\infty )\) is onetoone and onto, and its inverse is defined on \([0,+\infty ).\) A similar result holds for the function \( {M}^H_\varphi .\)
Proof
For every \(\xi \in {\mathbb {H}^n}\), \(r> 0\) and \(v \in V_1,\) with \(\Vert v\Vert =1\), set
The function \(\widehat{m}_\varphi ^H\) is continuous, and strictly increasing w.r.t. r, since \(\varphi \) is strictly Hconvex; thus,
Hence, by the Berge Maximum Theorem (see, for instance, [2]) \(m^H_\varphi \) is continuous, and
Let us show that the previous inequality is strict. The set \(\{v\in V_1:\ \Vert v\Vert =1\}\) is compact, and \(\widehat{m}_\varphi ^H(\xi , \cdot , \cdot )\) is continuous, then there exist v and \(v'\) such that \(\widehat{m}_\varphi ^H(\xi ,v,r)={m}^H_\varphi (\xi ,r)\) and \( \widehat{m}_\varphi ^H(\xi ,v',r')={m}^H_\varphi (\xi ,r')\). This implies that
Let us show that \(m^H_\varphi (\xi , \cdot )\) is unbounded, for every \(\xi .\) Suppose, by contradiction, that there exists \(L=L(\xi )>0\) such that \({m}^H_\varphi (\xi ,r)\le L\) for every \(r\ge 0.\) From the continuity of the function \(v\mapsto \widehat{m}^H_\varphi (\xi ,v,r),\) for every r there exists \(v_r\), with \(\Vert v_r\Vert =1\), such that \(m^H_\varphi (\xi ,r)= \widehat{m}_\varphi ^H(\xi ,v_r,r).\) Let \(r_n\rightarrow +\infty ;\) then, there exists \(\{v_{r_{n_k}}\}\) such that \(v_{r_{n_k}}\rightarrow \overline{v}.\) We have that
contradicting the assumption that \({m}^H_\varphi (\xi ,r)=\widehat{m}^H_\varphi (\xi ,v_r,r)\le L\) for every \(r>0.\) \(\square \)
4 Engulfing Property for HSections of HConvex Functions
This section is devoted to the study of the engulfing property E(H, K) for the Hsections of an Hconvex function. Our notion is different when compared with the one introduced by Capogna and Maldonado, and it generalizes the usual notion in the literature (see for example [19]); however, we will see that these notions are equivalent (see Proposition 4.2). In the second part of the section we prove that a sufficient condition for a function to satisfy the engulfing property E(H, K) is to have the round Hsections property, or, equivalently, the controlled Hslope (see Theorem 3.1). Finally, we will show, with an example, that the previous mentioned condition is only sufficient.
Let us start with our notion of engulfing for Hconvex functions defined in \(\mathbb {H}^n.\)
Definition 4.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. We say that \(\varphi \) satisfies the engulfing property E(H, K) (shortly, \(\varphi \in E(H,K)\)) if there exists \(K > 1\) such that, for any \(\xi \in {\mathbb {H}^n}\) and \(s > 0\), if \(\xi '\in S_\varphi ^H(\xi ,p,s)\) with \(p\in \partial _H \varphi (\xi )\), then
for every \(q\in \partial _H \varphi (\xi ')\).
As a matter of fact, as mentioned previously, in [13] a slightly different definition of engulfing is investigated in the framework of Carnot groups; if \(\mathbf{G} =\mathbb {H}^n,\) it can be stated as follows:
(we will refer to \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_K\) in case the constant K plays a role). Trivially, \(\varphi \in E(H,K)\) implies that \(\varphi \) satisfies \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_K\). The condition \(\mathrm {(}\mathbf {eng}_H\mathrm {)}\) is essentially onedimensional, as proved in the next
Proposition 4.1
(see [13]). Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be a strictly Hconvex and Hdifferentiable function. The function \(\varphi \) satisfies \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_K\) if and only if for every \(\xi \in {\mathbb {H}^n}\) and \(v\in V_1\) the function \(\varphi _{\xi ,v}:\mathbb {R}\rightarrow \mathbb {R}\) satisfies condition ii. in Theorem 2.1.
The following characterization provides an \({\mathbb {H}^n}\)version of the result in Theorem 2.1:
Proposition 4.2
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be a strictly Hconvex function. The following are equivalent:

i.
\(\varphi \) satisfies the engulfing property E(H, K), for some \(K>1;\)

ii.
\(\varphi \) satisfies condition \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_{K'},\) for some \(K'>1;\)

iii.
there exists a constant \(K''> 1\) such that, for any \(\xi \in {\mathbb {H}^n},\ \xi '\in H_\xi ,\) for any \(p\in \partial _H\varphi (\xi )\) and \(q\in \partial _H\varphi (\xi '),\)
$$\begin{aligned}&\frac{K''+1}{K''}\left( \varphi (\xi ')\varphi (\xi )p\cdot (\text {Pr}_1(\xi ')\text {Pr}_1(\xi ))\right) \\&\quad \le (qp)\cdot (\text {Pr}_1(\xi ')\text {Pr}_1(\xi ))\\&\quad \le (K''+1)\left( \varphi (\xi ')\varphi (\xi )p\cdot (\text {Pr}_1(\xi ')\text {Pr}_1(\xi ))\right) . \end{aligned}$$
In particular, if any of the conditions above holds, \(\varphi \) is Hdifferentiable.
Proof
Trivially, i. implies ii., and one can take \(K'=K.\) Let us show that ii. implies i. Let \(\xi '=\xi \circ \exp v\) be a point in \(S^H_\varphi (\xi , p,s),\) and consider the convex function \(\widehat{\varphi }_{\xi ,v}:\mathbb {R}\rightarrow \mathbb {R}\) defined in (3.3). Note that
and the function \(\widehat{\varphi }_{\xi ,v}\) satisfies condition ii. in Th. 2.1 with constant \(K'.\) From Theorem 1 in [11], \(\widehat{\varphi }_{\xi ,v}\in C^1(\mathbb {R}).\) Since this holds for every \(\xi , v,\) from Remark 3.1\(\varphi \) is Hdifferentiable everywhere and \(\partial _H\varphi (\xi )=\{\nabla _H\varphi (\xi )\}.\) Moreover, from Theorem 5.1 in [14], the function \(\widehat{\varphi }_{\xi ,v}\) satisfies the engulfing condition with constant \(2K'(K'+1).\) This is equivalent to say that
From (4.1), we get that
i.e., \(\varphi \) is in \(E(H,2K'(K'+1)).\)
In order to prove that ii. implies iii., let \(\xi '=\xi \circ \exp v\) and consider the convex function \(\widehat{\varphi }_{\xi ,v}.\) Note that \(p\cdot v\in \partial \widehat{\varphi }_{\xi ,v}(0)\) and \(q\cdot v\in \partial \widehat{\varphi }_{\xi ,v}(1).\) Then, by applying Proposition 2.1 in [11], we have that iii. holds with \(K''=K'.\) To conclude, let us show that iii. implies ii. Take \(\xi '=\xi \circ \exp v\in S^H_\varphi (\xi , p,s),\) where \(p\in \partial _H\varphi (\xi ),\) and let \(q\in \partial _H\varphi (\xi ').\) Then,
The second inequality in iii. gives
Then,
thus, \(\xi \in S^H_\varphi (\xi \circ \exp v,q,K''s),\) i.e., condition \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_{K''}\) is fulfilled. \(\square \)
Let us recall that a setvalued map \(T:\mathbb {H}^n\rightarrow \mathcal {P}(V_1)\) is said to be Hmonotone if, for all \(\xi \in \mathbb {H}^n,\) \(\xi '\in H_\xi ,\) \(p\in T(\xi ),\) \(q\in T(\xi '),\) then
(here \(V_1\cong \mathbb {R}^{2n})\) In particular, if \(\varphi \) is an Hconvex function, then the Hsubdifferential map \(\partial _H\varphi \) is an Hmonotone setvalued map (see [10]). The property iii. above requires, in fact, a stronger control on the Hmonotonicity, both from below and from above.
Let us now state the following crucial result, that provides a sufficient condition for E(H, K) via the round Hsections property; the relationship between round Hsections, or, equivalently, controlled Hslope, and the engulfing property corresponds to the similar one in \(\mathbb {R}^n\), for \(n\ge 2\):
Theorem 4.1
If \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) is an Hconvex function with round Hsections, then \(\varphi \) satisfies the engulfing property E(H, K), where K depends only on \(K_0\) in (3.4).
Proof
Since \(\varphi \) has round Hsections, Proposition 3.2 implies that \(\varphi \) is strictly Hconvex and Hdifferentiable. Let \(\xi '\in S_\varphi ^H(\xi ,s)\) be such that \(\xi '=\xi \circ \exp (r'v)\) for some v in \(V_1,\) with \(\Vert v\Vert =1\) and \(r'>0\); we will prove that \(\xi \in S_\varphi ^H(\xi ',Ks)\) where K depends only on \(K_0\) in (3.4).
Let R be such that
Since \(S_\varphi ^H(\xi ,s)\) is bounded, let us consider
Hence,
and \(0<r'\le r^\partial \). From the Hmonotonicity of the map \(\xi \mapsto \partial _H\varphi (\xi )\) we have that
Let us introduce the function \(\Phi :\mathbb {R}\rightarrow \mathbb {R}\) defined by
this function is strictly convex, with \(\Phi (0)=\Phi '(0)=0\). Let us consider the function \(\Pi :\mathbb {R}\rightarrow \mathbb {R}\) defined by
clearly, it represents the tangent to the graph of \(\Phi \) at \((r^\partial , \Phi (r^\partial ))\) with \(\Phi (r^\partial )>0\) and \(\Phi '(r^\partial )>0;\) hence we have
Since \(\varphi \) is Hconvex, the previous equalities and (3.2) give
The inequality above, together with (4.3) and (4.6), give
The Hconvexity of \(\varphi \) and (4.3) imply that \(\varphi _\xi (\xi \circ \exp (2r^\partial v))\le \varphi _\xi (\xi \circ \exp (2R v))\); hence we obtain
Let us consider \(\alpha \in \mathbb {N}\) such that \(K_0>2^{\alpha }.\) By iterating relation (3.5), we obtain
where C depends only on \(K_0\). The previous inequality, and relations (4.2) and (4.7), give
At this point, since \(S_\varphi ^H(\xi ,s)\) is open, there exists \(\tilde{\xi }=\xi \circ \exp (\tilde{r} v)\in S_\varphi ^H(\xi ,s)\) with \(\tilde{r}<0\). Taking into account that \(\xi '\in S_\varphi ^H(\xi ,s)\) and \(\tilde{\xi }\in S_\varphi ^H(\xi ,s)\), and using (4.8) and (4.2), we obtain
This implies that \(\tilde{\xi }\in S_\varphi ^H(\xi ',Ks)\), with \(K=1+ \frac{2C^{1+\alpha }}{K_0}\): since \(\xi \) belongs to the horizontal segment which joins \(\tilde{\xi }\) and \(\xi ',\) and since \(S_\varphi ^H(\xi ',Ks)\) is Hconvex, then \(\xi \in S_\varphi ^H(\xi ',Ks).\) By Proposition 4.2 the assertion is proved. \(\square \)
The following example is crucial in order to shed some light on the relationship between round sections and engulfing; indeed, it shows that the converse of the previous theorem fails. The idea is taken from an example due to Wang (see [24]) and set in \(\mathbb {R}^2;\) we adapt his idea to the case of the first Heisenberg group \(\mathbb {H}\).
Example 4.1
Consider the following differentiable and strictly convex function \(u:\mathbb {R}^2\rightarrow \mathbb {R},\)
The MongeAmpère measure \(\mu _u\) (we recall that \(\mu _u\) is defined by \(\mu _u(E)=\partial u(E)\) for every Borel set \(E\subset \mathbb {R}^2)\) is absolutely continuous with respect to the Lebesgue measure \(\cdot \), and it verifies the condition \(\mu _\infty \), i.e. for any \(\delta _1\in (0,1)\) there exists \(\delta _2\in (0,1)\) such that: for every section \(S_u(z,s)\), with \(z\in \mathbb {R}^2\), and for every Borel set \(B\subset S_u(z,s),\)
(see Definition 3.7 in [21]). This condition \(\mu _\infty \) is stronger than the (DC)doubling property (see, for example, relation (3.1.1) in [19]), i.e., there exist constants \(\alpha \in (0,1)\) and \(C>1\) such that
for every \(z,s>0\) (here \(\alpha S_{u}(z,s)\) denotes the open convex set obtained by \(\alpha \)contraction of \(S_{u}(z,\tau )\) with respect to its center of mass). In [20] and [17] it was shown that the (DC)doubling property of the measure \(\mu _{u}\) is equivalent to the engulfing property of the function u. Therefore, u satisfies the engulfing property.
Since the second derivative of u w.r.t. \(x_2\) is unbounded near the origin, so is \(\Vert D^2 u\Vert \); thus, u is not quasiuniformly convex (see Theorem 7.1i. and [21] for further details). However, a simpler argument can be advanced to prove that u is not quasiuniformly convex, that is one can show that u has not controlled slope in (2.12): in order to do that, we only remark that, taking into account that \(u(0,0)=0\) and \(\nabla u(0,0)=(0,0)\), we have, for large r,
Now let us consider the function \(\varphi :\mathbb {H}\rightarrow \mathbb {R}\) defined by \(\varphi (x,y,t)=u(x,y),\) for all \((x,y,t)\in \mathbb {H}.\) This function \(\varphi \) is \(\mathbb {R}^3\)convex, and hence Hconvex. Since
\(\varphi \) has not controlled Hslope, and hence has not round Hsections. However, since
it is easy to see that \(\varphi \) enjoys the engulfing property.
5 \(\pmb {{\mathbb {H}^n}}\)Sections of HConvex Functions and Their Engulfing Properties
In this section we will present our new definition of section in \(\mathbb {H}^n.\) First of all, we will prove that these \({\mathbb {H}^n}\)sections are fulldimensional, i.e., they contain a KorányiCygan ball. This allows to construct a topology in \(\mathbb {H}^n,\) as we will see in the next Sect. 6. In the second part, we introduce the condition of engulfing \(E({\mathbb {H}^n},K)\) for these new \({\mathbb {H}^n}\)sections. It will not be a surprise that \(\varphi \in E({\mathbb {H}^n},K)\) implies that \(\varphi \in E(H,K)\), while the converse implication is very hard and mysterious (at least to us). In order to shed some light on this, let us focus our attention on the functions having round Hsections, or, equivalently, controlled Hslope. As we will see, some technical estimates allow us to prove the first part of our main result in Theorem 1.1.
Let us start with our new notion of \({\mathbb {H}^n}\)section:
Definition 5.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function and let us fix \(\xi _0\in {\mathbb {H}^n}.\) For a given \(s>0,\) an \({\mathbb {H}^n}\)section of \(\varphi \) at height s, with \(p_0\in \partial _H \varphi (\xi _0)\), is the set
In case \(\varphi \) is Hdifferentiable at \(\xi _0\), we will denote the \({\mathbb {H}^n}\)section at \(\xi _0\) with height s by \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi _0,s)\), for short.
Let us spend a few words on the definition above. Lemma 1.40 in the fundamental book by Folland and Stein [16] guarantees that, in every stratified group \((\mathbf{G} ,\circ )\) with homogeneous norm \(\Vert \cdot \Vert _\mathbf{G} \), there exists a constant \(C>0\) and an integer \(k\in \mathbb {N}\) such that any \(\xi \in \mathbf{G} \) can be expressed as \(\xi =\xi _1\circ \xi _2\circ \ldots \circ \xi _k\), with \(\xi _i\in \exp (V_1)\) and \(\Vert \xi _i\Vert _\mathbf{G} \le C \Vert \xi \Vert _\mathbf{G} ,\) for every i. If \(\mathbf{G} =\mathbb {H}^n,\) the mentioned k is exactly 3, for every \(n\ge 1.\) In other words, every point \(\xi \in {\mathbb {H}^n}\) can be reached from the origin e following a path of three consecutive horizontal segments. The idea behind Definition 5.1 takes inspiration from this result, in view of providing a family of sets with nonempty interior. Let us define, for every \(\xi \in {\mathbb {H}^n}\) and \(r>0\),
Clearly, \(\delta _\lambda \left( \widetilde{B}(e,r)\right) =\widetilde{B}(e,\lambda r),\) and the associated distance \(\widetilde{d}\) in \({\mathbb {H}^n}\) is leftinvariant and homogeneous; hence, it is biLipschitz equivalent to \(d_g\) and to any other leftinvariant and homogeneous distance in \({\mathbb {H}^n}\). Moreover, due to the Folland–Stein Lemma, we have that, for every \(\xi \in {\mathbb {H}^n}\) and \(r>0\),
where C is the constant in the mentioned lemma.
Let us prove the first fundamental property of the \({\mathbb {H}^n}\)sections, namely \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi _0,p_0,s)\) is fulldimensional.
Proposition 5.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. Then, for every \(\xi _0\in {\mathbb {H}^n},\ p_0\in \partial _H \varphi (\xi _0)\) and \(s>0\), there exists \(r>0\) such that
Proof
Without loss of generality, we set \(\xi _0=e\). Since the Hsubdifferential map \(\partial _H\varphi \) brings compact sets into compact sets (see, for instance, Proposition 2.1 in [4]), there exists a positive constant R such that
Moreover, since \(\varphi \) is locally Lipschitz (see Theorem 1.2 in [5]), there exists a positive constant L such that
Set
where C is the constant in the Folland–Stein Lemma. We will show that \(B_g(e,r)\subset \mathbb {S}_\varphi ^{\mathbb {H}^n}(e,p_0,s).\)
Take any \(\xi \in B_g(e,r)\); then, \(\xi =\exp (v_1)\circ \exp (v_2)\circ \exp (v_3)\) for suitable \(\{v_i\}_{i=1}^3\subset V_1\) such that \(\Vert v_i\Vert \le C \Vert \xi \Vert _g,\) for \(i=1,2,3.\) Set
Note that \(\xi _i\in \widetilde{B}(e,r)\) for \(i=1,2,3.\) Then, for every \(p_i\in \partial _H\varphi (\xi _i)\) (\(i=1,2\)), by (5.4) and (5.5) we have
and
Since \(\varphi (\xi _1)\varphi (e)p_0\cdot v_1 \le (L+R)\,\Vert v_1\Vert < s, \) from (5.6) and (5.7) we get the claim. \(\square \)
Starting from these \(\mathbb {H}^n\)sections, we introduce the following engulfing property:
Definition 5.2
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. We say that \(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n},K)\) if there exists \(K > 1\) such that, for any \(\xi \in {\mathbb {H}^n},\ p\in \partial _H \varphi (\xi )\) and \(s > 0\), if \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,p,s)\), then
for every \(q\in \partial _H \varphi (\xi ')\).
The engulfing property E(H, K) is related to this engulfing property \(E(\mathbb {H}^n,K)\) as well as condition \(\mathrm {(}\mathbf {eng}_H\mathrm {)}\) is related to the following condition:
We will refer to \(\mathrm {(}\mathbf {eng}_{\mathbb {H}^n}\mathrm {)}_K\) in case we need to specify the constant K in the previous condition.
It is clear that
Remark 5.1
If \(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n},K)\), then condition \(\mathrm {(}\mathbf {eng}_{\mathbb {H}^n}\mathrm {)}_K\) holds.
The converse of the previous remark is a delicate question: the aim of this section is, essentially, to prove that, under further conditions on \(\varphi \), the converse of Remark 5.1 holds.
The relationship between conditions \(\mathrm {(}\mathbf {eng}_H\mathrm {)}\) and \(\mathrm {(}\mathbf {eng}_{\mathbb {H}^n}\mathrm {)}\) is the following:
Proposition 5.2
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function. Then \(\varphi \) satisfies condition \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_K\) if and only if \(\varphi \) satisfies condition \(\mathrm {(}\mathbf {eng}_{\mathbb {H}^n}\mathrm {)}_K\).
Proof
If \(\varphi \) satisfies \(\mathrm {(}\mathbf {eng}_{\mathbb {H}^n}\mathrm {)}_K\), it is clear that \(\mathrm {(}\mathbf {eng}_H\mathrm {)}_K\) holds. Let us prove the converse. Take any \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,p,s)\), i.e. \(\xi '=\xi \circ \exp (v_1)\circ \exp (v_2)\circ \exp (v_3),\) with \(v_i\in V_1\) and with
we have to show that \(\xi \in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',q,K's)\), for every \(q\in \partial _H \varphi (\xi ')\). The assumption implies
Hence, for every \(q\in \partial _H \varphi (\xi ')\),
\(\square \)
Clearly, if \(\varphi \) is a strictly Hconvex function satisfying the engulfing property \(E({\mathbb {H}^n},K)\), then Remark 5.1, Proposition 5.2 and Proposition 4.2 imply that \(\varphi \) is Hdifferentiable.
The next result will be crucial to our purposes:
Proposition 5.3
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hdifferentiable and strictly Hconvex function. Then, for every \( \xi \in {\mathbb {H}^n},\ r>0\), we have
Let us emphasize that, despite its appearance, the first set in (5.8) is not an \({\mathbb {H}^n}\)section, since \(m_\varphi ^H(\xi _1,r)\), for \(\xi _1\in S^H_\varphi (\xi , m_\varphi ^H(\xi ,r))\), and \(m_\varphi ^H(\xi _2,r)\), for \(\xi _2\in S^H_\varphi (\xi _1, m_\varphi ^H(\xi _1,r))\), are not fixed values. A similar comment holds for the set in (5.9).
Proof of Proposition 5.3
By the inclusions in (3.6), we easily have
for every \( \xi \in {\mathbb {H}^n},\ r>0\). Hence the assertion holds. \(\square \)
In order to prove our main result concerning the engulfing property of the \(\mathbb {H}^n\)sections, an extension to the Heisenberg case of the inequalities (2.8), (2.9) and (2.11) turns out to be quite useful:
Proposition 5.4
Let \(\varphi \) be a strictly Hconvex function in E(H, K). Then, for every \(r\ge 0\) and \(\xi \in {\mathbb {H}^n},\) we have
where \( B_1,\ B_2\) and \(B_4\) depend only on K, and \(B_i>1\).
Proof
Proposition 4.2 implies that \(\varphi \) is Hdifferentiable and, if we consider its restriction to any horizontal segment, we obtain a strictly convex and differentiable function. To be precise, for every \(\xi \in {\mathbb {H}^n}\) and \(v\in V_1\) with \(\Vert v\Vert =1\) the function \(\widehat{\varphi }_{\xi ,v}:\mathbb {R}\rightarrow \mathbb {R}\), defined as in (3.3), satisfies condition ii. in Th. 2.1. By (2.8) in Remark 2.1 we obtain
where \(B_1\) depends only on K. Hence we have
taking the maximum w.r.t. to v, with \(\Vert v\Vert =1,\) we obtain (5.10).
A similar proof, via inequality (2.9) in Remark 2.1 and inequality (2.11) in Proposition 2.1, shows (5.11) and (5.12), respectively. \(\square \)
In the final part of this section we will prove our main result concerning the relationship between round Hsections and the engulfing property of the \(\mathbb {H}^n\)sections. The proof will be quite technical, deserving a few previous estimates.
Let \(\varphi :\mathbb {H}^n\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections (with constant \(K_0\)). Then, \(\varphi \in E(H,K)\), and has controlled Hslope (with constant \(K_1\)), where both \(K,K_1\) depend on \(K_0.\) Denote by \(\gamma \) any positive integer such that
Thus, from (3.7), and by iterating inequality (5.12), we obtain
Then, we have that
for every \(r>0\) and \(\xi \in {\mathbb {H}^n}\), where \(\gamma >1\) depends only on \(K_0\) in (3.4).
The next proposition holds:
Proposition 5.5
Let \(\varphi :\mathbb {H}^n \rightarrow \mathbb {R}\) be a function with round Hsections (with \(K_0\) as in (3.4)). Then, there exists a constant \(C_1>0\) such that, if \(\xi '\in S_\varphi ^H (\xi ,s)\), then
for r such that \(s= m^H_\varphi (\xi ,r)\). The constant \(C_1\) depends only on \(K_0\).
Proof
Since \(\varphi \) has round Hsections, it is strictly Hconvex, Hdifferentiable and it satisfies the engulfing property E(H, K), where K depends only on \(K_0\). Let \(\xi '=\xi \circ \exp v\in S_\varphi ^H (\xi ,s),\) and set r such that \(s= m^H_\varphi (\xi ,r)\): clearly,
Moreover, since \(\xi '\in S_\varphi ^H (\xi , m^H_\varphi (\xi ,r))\), by (3.6) we have that
Furthermore, since \(\varphi \in E(H,K)\), we have that \(\xi '\in S_\varphi ^H (\xi ,{M}^H_\varphi (\xi ,2r))\) gives
This implies \(\xi '\circ \exp (\pm rv/\Vert v\Vert )\in S_\varphi ^H (\xi ',K{M}^H_\varphi (\xi ,2r))\). Now, by (5.10) and (3.4), we have
where \(B_1\) depends only on \(K_0\). Then,
\(\square \)
A result similar to Proposition 5.5, involving now the \({\mathbb {H}^n}\)sections \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\), holds, but the proof is much more delicate:
Proposition 5.6
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be a function with round Hsections (with \(K_0\) as in (3.4)). Then, there exists a constant \(B_5>0\) such that, if \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\), then
for r such that \(s= m^H_\varphi (\xi ,r)\). The constant \(B_5\) depends only on \(K_0\).
Proof
Since \(\varphi \) has round Hsections, it belongs to E(H, K), where K depends only on \(K_0\), and for every \(r\ge 0\) and \(\xi \in {\mathbb {H}^n}\) the inequality (5.12) holds. In addition, by Proposition 3.3, the function \(m_\varphi ^H(\xi ,\cdot ):[0,+\infty )\rightarrow [0,+\infty )\) is invertible.
Take any \(\xi _3\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi _0,s)\), i.e. \(\xi _1\in S_\varphi ^H (\xi _0,s)\), \(\xi _2\in S_\varphi ^H (\xi _1,s)\) and \(\xi _3\in S_\varphi ^H (\xi _2,s)\), with s such that \(s=m^H_\varphi (\xi _0,r)\). By Proposition 5.5 and \(\xi _1\in S_\varphi ^H (\xi _0,s)\), we have
Similarly, since \(\xi _2\in S_\varphi ^H (\xi _1,s)\), we have
Let us prove that there exists a constant C, which depends only on \(C_1\) and \(B_4,\) and hence on \(K_0,\) such that
Inequality (5.12) is equivalent to
by choosing \(\beta \in \mathbb {N}\) such that \(C_1\le B_4^\beta \), iterating the previous inequality and taking into account that \(\tilde{s}\mapsto \left( m_\varphi ^H(\xi ,\cdot )\right) ^{1}(\tilde{s})\) is an increasing function we obtain
Hence, (5.18) holds with \(C= 2^\beta \); now, by (5.16), (5.19) and (5.17), we obtain
A similar argument proves that \(\xi _3\in S_\varphi ^H (\xi _2,s)\) implies
Now, recalling that \(s= m^H_\varphi (\xi _0,r)\), the previous inequality gives
Finally, (5.11) and (5.20) implies
and the proof in finished. \(\square \)
In order to introduce and prove the main result of the section, we need the following
Lemma 5.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections.

a.
If \(\xi '\in S_\varphi ^H (\xi ,m_\varphi ^H(\xi ,r))\) for some \(r>0\), then
$$\begin{aligned} m_\varphi ^H(\xi ,r)\le m_\varphi ^H(\xi ',2^{1+\gamma }r), \end{aligned}$$(5.21)with \(\gamma \) as in (5.13);

b.
if \(\xi '\in S_\varphi ^H (\xi , M_\varphi ^H(\xi ,r))\) for some \(r>0\), then
$$\begin{aligned} M_\varphi ^H(\xi ',r)\le m_\varphi ^H(\xi ,2^{2+3\gamma +\tilde{\gamma }}r), \end{aligned}$$(5.22)with \(\tilde{\gamma }\) as in (5.27) which depends only on \(K_0\).
Proof
Let us consider \( \xi '=\xi \circ \exp v\in S_\varphi ^H (\xi ,m_\varphi ^H(\xi ,r))\): by (3.6) we have
The Hconvexity of \(\varphi \) and the Hmonotonicity of \(\nabla _H\varphi \) give
Again the Hconvexity of \(\varphi \) and (5.23)(5.25) give
Therefore (5.21) follows from (5.14).
Let us prove b. Take any \(\xi '\in S_\varphi ^H (\xi , M_\varphi ^H(\xi ,r))\); since \(\varphi \in E(H,K),\) with K depending on \(K_0\) only (see Theorem 4.1), then \(\xi \in S_\varphi ^H (\xi ',K M_\varphi ^H(\xi ,r))\). From (5.14) we have \( \xi '\in S_\varphi ^H (\xi , M_\varphi ^H(\xi ,r))\subset S_\varphi ^H (\xi , m_\varphi ^H(\xi ,2^\gamma r)),\) and (5.21) implies that
Now, let \(\tilde{\gamma }\in \mathbb {N}\) be such that
By iterating inequality (5.12) and (5.14), inequality (5.26) gives
Hence, \(\xi \in S_\varphi ^H (\xi ',K M_\varphi ^H(\xi ,r))\subset S_\varphi ^H (\xi ',m_\varphi ^H(\xi ',2^{1+2\gamma +\tilde{\gamma }}r)).\) Finally, the inequalities (5.14) and (5.21) imply
\(\square \)
We are now in the position to prove the first part of our main result in Theorem 1.1:
Proof of Theorem 1.1 i
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections. Let us prove that \(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n}, K)\). Fix \(\xi \in {\mathbb {H}^n}\) and \(s>0\). Let us suppose that \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\): we have to prove that \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\subset \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',Ks)\), where K is a constant which depends only on \(K_0\) in (3.4).
Let r be such that \(s=m_\varphi ^H(\xi ,r)\). By definition,
For every \(\xi _1\in S_\varphi ^H (\xi ,m_\varphi ^H(\xi ,r)),\) by applying a. in Lemma 5.1, we get
and, by (5.29), we get
For every \( \xi _2\in S_\varphi ^H (\xi _1,m_\varphi ^H(\xi _1,2^{1+\gamma } r))\) with \(\xi _1\in S^H_\varphi (\xi , m_\varphi ^H(\xi ,r)),\) via a. in Lemma 5.1 we get
Using now (5.30) and (5.32), relation (5.31) becomes
Since \(\gamma >0\), we have the following inclusions:
where the last inclusion comes from (5.8). Then, using the inclusions in (5.3), we get
where C is the constant in the Folland–Stein Lemma. Inclusions (5.9) and (5.34) give
Now, applying twice (5.22) in Lemma 5.1, we have, by the previous inclusion,
Set \(\tilde{C}=3C2^{7+8\gamma +2\tilde{\gamma }},\) and take any \(\delta \in \mathbb {N}\) such that \(\tilde{C}\le 2^\delta \); clearly, both \(\tilde{C}\) and \(\delta \) they depend only on \(K_0\). Hence, we have the following inclusions:
\(\square \)
6 Balls and Quasidistances via the \({\mathbb {H}^n}\)Sections of HConvex Functions
It is known that there is a deep connection between the existence of a quasidistance d on a given set X and the existence of a family of subsets \(\{S(x,s)\}_{\{x\in X,\, s>0\}}\) enjoying the following properties
 \((P_1)\):

\(\bigcap _{s>0} S(x,s)=\{x\},\) for every \(x\in X\);
 \((P_2)\):

\(\bigcup _{s>0} S(x,s)=X,\) for every \(x\in X\);
 \((P_3)\):

for each \(x\in X\), \(s\mapsto S(x,s)\) is a non decreasing map;
 \((P_4)\):

there exists a constant H such that, for all \(y\in S(x,s)\),
$$\begin{aligned} S(x,s)\subset & {} S(y,Hs), \end{aligned}$$(6.1)$$\begin{aligned} S(y,s)\subset & {} S(x,Hs). \end{aligned}$$(6.2)
As a matter of fact, the following result holds:
Lemma 6.1
(see Lemma 1 in [1]) Let X be a set and \(S: X\times \mathbb {R}^+\rightarrow {\mathcal P}(X)\) be a setvalued map such that the family \(\{S(x,s)\}\) has the properties \((P_1)\)\((P_4)\). Then, the function \(d:X\times X\rightarrow [0,+\infty )\) defined by
is a quasidistance. On the other hand, given a quasidistance d defined on X, the family of the dballs in X satisfies the properties \((P_1)\)\((P_4)\).
In particular, in [1] the authors prove that the sections \(S_u(x,r)\) of a convex function \(u:\mathbb {R}^k\rightarrow \mathbb {R}\) satisfying the engulfing property, generate a quasidistance.
Let us now consider an Hconvex function \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) with round Hsections; by taking all \(s>0\) and \(\xi \in {\mathbb {H}^n}=\mathbb {R}^{2n+1}\) we obtain a family of sets \(\{\mathbb {S}_\varphi ^\mathbb {H}\left( \xi ,s \right) \}_{\{\xi \in {\mathbb {H}^n},\, s>0\}}\) (the \({\mathbb {H}^n}\)sections) for which conditions \((P_1)\)\((P_3)\) trivially hold; moreover, due to Theorem 1.1, such family satisfies the engulfing property \(E({\mathbb {H}^n},K)\), i.e. condition (6.1).
The next result shows that the family of \({\mathbb {H}^n}\)sections satisfies condition (6.2) too:
Theorem 6.1
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections. Then, there exists a constant \(\tilde{K},\) which depends only on \(K_0,\) such that, if \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\), then \(\mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',s)\subset \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,\tilde{K} s)\).
Proof
The proof follows the ideas in the proof of Theorem 1.1. Fix \(\xi \in {\mathbb {H}^n}\ s>0\) and \(\xi '\in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ,s)\); let r be such that \(s=m_\varphi ^H(\xi ',r)\). Theorem 1.1 guarantees that \(\varphi \) satisfies the engulfing property \(E({\mathbb {H}^n}, K)\), where K depends only on \(K_0\). Hence, \(\xi \in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',Ks)\). Proposition 5.6 implies that
for \(\widehat{r}\) such that \(Ks= m^H_\varphi (\xi ',\widehat{r})\) (the constant \(B_5\) depends only on \(K_0\)). Since, by Proposition 3.3, the function \(r\mapsto m^H_\varphi (\xi , r)\) is an increasing function, we obtain
By definition,
Using exactly the same arguments as in the proof of Theorem 1.1, that allow us to pass from (5.29) to (5.33) (essentially, by exchanging the role of \(\xi \) and \(\xi '\)), we obtain
Now, taking into account the definition of \(\tilde{\gamma }\) in (5.27) and iterating inequality (5.12), we get
Since \(\xi \in \mathbb {S}_\varphi ^{\mathbb {H}^n}(\xi ',Ks)\), we obtain
Using exactly the same arguments that allow us to pass from (6.4) to (6.5) (essentially, by exchanging the role of r with \(2^{\tilde{\gamma }}r\)), we obtain
Now, taking into account that \(\tilde{\gamma }>0\), relations (6.5) and (6.8) give
where C in the previous inclusions is the constant in the Folland–Stein Lemma. Using the same arguments that allow us to pass from (5.34) to (5.35) (essentially, by replacing \(3C2^{3+2\gamma }r\) with \(3C2^{3+2\gamma +\tilde{\gamma }}r\)), we obtain
Set \(\widehat{C}=3C2^{7+8\gamma +3\tilde{\gamma }}\) and take \(\widehat{\delta }\in \mathbb {N}\) such that \(\widehat{C}\le 2^{\widehat{\delta }}\); clearly, \(\widehat{C}\) and \(\widehat{\delta }\) depend only on \(K_0\). We have the following inclusions:
which concludes the proof. \(\square \)
We are now in the position to prove the second part of our main result in Theorem 1.1:
Proof of Theorem 1.1 ii
Let \(\varphi :{\mathbb {H}^n}\rightarrow \mathbb {R}\) be an Hconvex function with round Hsections. The previous arguments, together with Lemma 6.1 and Lemma 2 in [1], give that
is a quasidistance in \({\mathbb {H}^n}\). Moreover, if \(B_\varphi (\xi ,r)\) denotes the \(d_\varphi \)ball of center \(\xi \in {\mathbb {H}^n}\) and radius \(r>0\), we have that there exists H which depends only on \(K_0\) in (3.4) such that
\(\square \)
The definition of \(\mathbb {H}^n\)sections via subsequent constructions of Hsections makes hard its description in terms of functional inequalities. However, in the very simple case of the function \(\varphi :\mathbb {H}\rightarrow \mathbb {R}\) defined by \(\varphi (x,y,t)=x^2+y^2,\) we are able to fully describe the set \(\mathbb {S}_\varphi ^\mathbb {H}(e, r)\) by providing explicitly the equation of its boundary. While in the Euclidean case the function \(u(x)=\Vert x\Vert ^2,\) with \(x\in \mathbb {R}^n\), gives rise to the sections \(S_u(x_0,s)=B^{\mathbb {R}^n}(x_0,\sqrt{s}),\) i.e., the usual balls in \(\mathbb {R}^n,\) in the case of the first Heisenberg group \(\mathbb {H}\) and with the mentioned function \(\varphi \) we obtain \(S_\varphi ^\mathbb {H}(\xi _0,s)=\widetilde{B}(\xi _0,\sqrt{s}),\) and the family of \(\mathbb {H}\)sections of \(\varphi \) consists of the \(\widetilde{B}\)balls in (5.2).
Example 6.1
Let us consider \(\varphi :\mathbb {H}\rightarrow \mathbb {R}\) defined by \(\varphi (x,y,t)=x^2+y^2\). This function is \(\mathbb {R}^3\)convex, and hence Hconvex. Since \(\partial _H\varphi (x,y,t)=\{2(x,y)\}\), the horizontal section \(S^H_\varphi (\xi _0,s)\) is given by
for \(\xi _0=(x_0,y_0,t_0)\) and \(s>0\). Hence, for this particular \(\varphi ,\) we have that
and, therefore,
Since, from the definition of \(\mathbb {H}\)section, \(\mathbb {S}^\mathbb {H}_\varphi (\xi _0,s)=\xi _0\circ \mathbb {S}^\mathbb {H}_\varphi (e,s)\), we will focus on the particular case \(\xi _0=e.\) We claim that, for every \(r>0\),
Let us try to give the idea of its construction. Fix \(r>0\). First of all, note that

\(\widetilde{B}(e,\sqrt{r})\) is radial with respect to the taxis;

\(\widetilde{B}(e,\sqrt{r})\) is symmetric with respect to the xyplane.
In particular, it is sufficient to identify the points of the set \(\partial \widetilde{B}(e,\sqrt{r})\) in \(\mathbb {H}\cap \{t\ge 0\}.\) To this purpose, for every \(\theta \in [2\pi /3,0]\) let us consider the points
Trivially, \(\eta ^0=(3\sqrt{r},0,0)\in \partial \widetilde{B}(e,\sqrt{r})\). Let us motivate our choice in (6.13). Let \(v_i\in V_1\cong \mathbb {R}^2\), for \(i=1,2,3\), and consider the point
we have \((x,y)=v_1+v_2+v_3,\) and t/4 is equal to the area of the polygon \(P={\mathrm {co}}\{ (0,0),\ v_1,\ v_1+v_2,\ v_1+v_2+v_3\}\subset \mathbb {R}^2,\) where “co” denotes the convex hull (for details on this application of Stokes’ Theorem, see, for example, Section 2.3 in [12]). In order to construct \(\partial \widetilde{B}(e,\sqrt{r})\cap \{(x,y,t)\in \mathbb {H}:\ t\ge 0\}\) we restrict our attention to the points \(\eta \) in (6.14) with the following features:

\(\Vert v_i\Vert =\sqrt{r};\)

the angles \(\widehat{v_1,v_2}\) and \(\widehat{v_2,v_3}\) are equal to \(\theta \)
(this choice will be explained later on). Due to the symmetries of \(\widetilde{B}(e,\sqrt{r})\), we set
With this choice, from (6.13) one simply gets that
Clearly, \(t(\theta )\ge 0\) for \(\theta \in [\pi ,0]\). In the case \(\theta =2\pi /3\), P turns out to be an equilateral triangle, and \(\eta ^{2\pi /3}=(0,0,\sqrt{3} r)\); in the case \(\theta \in [\pi ,2\pi /3)\), we have that \(\eta ^{\theta }\) is an interior point of \(\widetilde{B}(e,\sqrt{r})\). Therefore, we restrict our attention to the points \(\eta ^\theta \) as in (6.13). Simple computations give that, for \(\theta \in [2\pi /3,0],\)
Note that, if \(\theta =\pi /3,\) the function \(t(\theta )\) reaches its maximum \( 3\sqrt{3} r\) and, in this case, \(d(\pi /3)=2\sqrt{r}\). Consider the change of variable \(z=\sqrt{r} (1+2\cos \theta );\) due to the symmetry of \(\widetilde{B}(e,\sqrt{r}),\) we obtain that
and thus we get the expression in (6.12).
Finally, let us explain briefly the restrictions imposed in (6.15) to obtain (6.12).
First, it is easy to see that, if in (6.14) we set \(\Vert v_i\Vert =\sqrt{r}'\), with \(0<r'<r\), we obtain that \(\eta \) in (6.14) is in \(\partial \widetilde{B}(e,\sqrt{r'})\subset \widetilde{B}(e,\sqrt{r})\); a similar argument holds for \(\eta \) in (6.14), with the choice \(\Vert v_i\Vert <\sqrt{r}\).
Secondly, let us motivate the restriction \(\widehat{v_1,v_2}=\widehat{v_2,v_3}=\theta \) in (6.16). Fix \(\theta \in (2\pi /3,0)\), consider \(v_i\) as in (6.15) and the mentioned polygon P; using (6.16), the area of P is exactly \(\sin \theta (1+\cos \theta ).\) If one looks for the triplet of vectors \(v_i\), with \(\Vert v_i\Vert =\sqrt{r}\) for \(i=1,2,3,\) such that \(v_1+v_2+v_3=(x(\theta ),y(\theta ))\) and such that the area of the associated polygon P is the biggest one, then one obtains exactly the vectors \(v_i\) in (6.15). This proves that \(\eta ^\theta \) belongs to the boundary of our \(\mathbb {H}\)section. We leave the details and their tedious calculations to the interested reader.
7 Final Remarks and Open Questions
Question 1
The assumption of the round Hsections property for an Hconvex function \(\varphi \) is a sufficient condition in order to guarantee that \(\varphi \) satisfies the engulfing property \(E(\mathbb {H}^n,K).\) It would be nice to weaken this assumption and prove that a function with the engulfing property E(H, K) satisfies the engulfing property \(E({\mathbb {H}^n},K)\).
Question 2
In [13] the authors study the engulfing property for convex functions in a generic Carnot group \(\mathbf{G} \); as a matter of fact, in this more general framework, the related definition of \(\mathbf{G} \)sections (as in Definition 5.1) would be affected by the different geometry of the group \(\mathbf{G} \), by the number of the steps and, especially, by the number of consecutive horizontal segments needed to connect any pair of points. Moreover, in a Carnot group with step greater than 2, a socalled horizontal line, i.e., a set \(\{\xi \circ \exp sv\}_{s\in \mathbb {R}},\) is not a line in the Euclidean sense, as well as a horizontal plane is not a hyperplane in the Euclidean sense. This leads us to think that the \(\mathbf{G} \)sections may have a very peculiar shape.
Question 3
By Theorem 3.3.10 in [19], the engulfing property for a convex function \(\varphi :\mathbb {R}^n\rightarrow \mathbb {R}\) implies the existence of \(C > 0\) and \(p\ge 1\) such that for every \(0< r < s\le 1,\) \(x_0\in \mathbb {R}^n,\) \(t > 0,\) and \(x\in S_\varphi (x_0,rt)\) we have the inclusion
note that, under the assumptions above, the function \(\varphi \) is differentiable (see [11]). Under any suitable version of the engulfing property in \(\mathbb {H}^n,\) can a similar inclusion be proved in \(\mathbb {H}^n?\)
Question 4
In [21] the authors prove, among other things, that the notion of round sections in Definition 2.1, controlled slope in (2.12), quasi uniform convexity, and quasiconformity are strictly related properties. To be precise, the next result holds (see Theorem 3.1 in [21]):
Theorem 7.1
Let \(n\ge 2\), and let \(u:\mathbb {R}^n\rightarrow \mathbb {R}\) be a convex function. The following are equivalent:

i.
u is quasiuniformly convex function, i.e. u is not affine, \(u\in W_{loc}^{2,n}\) and there exists a constant \(K\ge 1\) such that
$$\begin{aligned} \Vert \nabla ^2 u (x)\Vert ^n\le K \text {det} \nabla ^2 u(x),\qquad \mathrm {a.e.}\, x\in \mathbb {R}^n; \end{aligned}$$(7.1) 
ii.
u is differentiable and \(\nabla u:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is quasiconformal, recalling that an injective map \(F:\mathbb {R}^n\rightarrow \mathbb {R}^n\) is quasiconformal if \(F\in W_{loc}^{1,n}\) and there exists a constant \(K\ge 1\) such that
$$\begin{aligned} \Vert \nabla F (x)\Vert ^n\le K \text {det} \nabla F (x),\qquad \mathrm {a.e.}\, x\in \mathbb {R}^n; \end{aligned}$$(7.2) 
iii.
u is differentiable, but not affine, and has controlled slope;

iv.
u has round sections.
On the other hand, it is well known that the notion of quasiconformal map on \({\mathbb {H}^n}\) has been introduced and intensively studied (see for example [12]). In this paper we introduce the notion of Hcontrolled slope and round Hsections for an Hconvex function but, at least to our knowledge, a horizontal notion of quasiuniform convexity for Hconvex function does not exist in the literature. Our future aim will be to investigate a horizontal version of Theorem 7.1.
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Calogero, A., Pini, R. The Engulfing Property for Sections of Convex Functions on the Heisenberg Group and the Associated Quasidistance. J Geom Anal 31, 10336–10373 (2021). https://doi.org/10.1007/s12220021006487
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DOI: https://doi.org/10.1007/s12220021006487
Keywords
 Heisenberg group
 Hconvex function
 Section of Hconvex function
 Engulfing property
 Quasidistance
 Round Hsections