The Engulfing Property for Sections of Convex Functions on the Heisenberg Group and the Associated Quasi-distance

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}^n}$$\end{document}. 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 Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}^n}$$\end{document}-section, as well as a new condition of engulfing associated to the Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}^n}$$\end{document}-sections, for an H-convex function defined in Hn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^n.$$\end{document} These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}^n}$$\end{document}-sections, with their engulfing property, will lead to the definition of a quasi-distance in Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}^n}$$\end{document} 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 H-sections for an H-convex function, and by its connection with the engulfing properties.


Introduction
Given a convex function u : R n → R, for every x 0 ∈ R n , p ∈ ∂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 ∈ E(R n , K )) if there exists K > 1 such that for any x ∈ R n , p ∈ ∂u(x), and s > 0, if y ∈ S u (x, p, s), then S u (x, p, s) ⊂ S u (y, q, K s), for every q ∈ ∂u(y). The functions u in the class E(R n , K ) have been studied in connection with the solution to the Monge-Ampère equation det D 2 u = μ, where μ is a Borel measure on R n . In this framework, a C 1, β -estimate for the strictly convex, generalized solutions to the Monge-Ampère equation was proved by Caffarelli ( [7,8]), under the assumption that the measure μ 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∈R n , s>0} , in case u is a convex differentiable function in E(R n , K ). In [1], it is shown that, in this case, one can define a quasi-distance d on R n as follows: d(x, y) := inf {s > 0 : x ∈ S u (y, s), y ∈ S u (x, s)} .
(1. 2) In addition, if B d (x, r ) is a d-ball of center x and radius r , then In the archetypal case u(x) = x 2 , with x ∈ R n , one has S u (x, s) = B R n (x, √ s), and hence the family of sections of u consists of the usual balls in R n .
In the case of convex functions defined in a Carnot group G, in [13] Capogna and Maldonado introduced some appropriate geometric objects, that can be considered as the sub-Riemmannian analogue of the classical sections, as well as a naturally related notion of horizontal engulfing. Given a horizontally convex function ϕ : G → R, ξ 0 ∈ G, p ∈ R m 1 , s > 0, the section S H u (ξ 0 , p, s) (H -sections, from now on, where H stands for horizontal) is defined as follows: (1.4) where V 1 ∼ = R m 1 is the first layer of the stratification of the Lie algebra of G; in case ϕ is horizontally differentiable at ξ 0 , we will denote such H -section by S H ϕ (ξ 0 , s), for short. The mentioned authors say that a horizontally convex and differentiable function ϕ satisfies the engulfing property if there exists K > 1 such that, for every ξ, ξ ∈ G and s > 0, if ξ ∈ S H ϕ (ξ, s), then ξ ∈ S H ϕ (ξ , K s). Let us stress that the definition of H -section in (1.4) and the notion of engulfing are affected by the sub-Riemannian 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 is a quasi-distance in H n ; moreover, for the d ϕ -balls, an H n -version of the inclusions in (1.3) holds true (see (6.10)

below).
Here, the archetypal example in H of the H -convex function ϕ(x, y, t) = x 2 + y 2 gives S H ϕ (ξ, s) = B(ξ, √ s), that is, the family of H-sections of ϕ consists of the B-balls of a left-invariant and homogeneous distance d (see (5.2) and Example 6.1).
The property of round H -sections is actually stronger than the horizontal engulfing; we are able to provide an example of an H -convex function which satisfies the horizontal engulfing property but does not have round H -sections, 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 full-dimensional 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 R n , together with the structure of H n and the notion of horizontal convexity. In Sect. 3 we introduce the H -sections, and we show that round H -sections and controlled H -slope are equivalent property for these H -sections (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 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.

Preliminary Notions and Results
In the paper, we will deal with H -convex functions defined on the Heisenberg group H n . As we will see later, the notion of H -convexity requires that, for every point ξ ∈ H n , one looks at the behaviour of the function under two points of view. The first one is one-dimensional, since the restriction of the function to any horizontal line {ξ • exp tv} t∈R , with v ∈ V 1 , is an ordinary convex function; the second one is 2ndimensional, according to the fact that v ∈ V 1 ∼ = R 2n , or, equivalently, the horizontal lines through ξ span the 2n-dimensional horizontal plane H ξ . For these reasons, the first part of this section will be devoted to some results related to the engulfing property of convex functions u : R n → R, both in the case n = 1, and in the case n ≥ 2. In the second part we will recall the notion of H -convexity, together with some related results, for functions defined on the Heisenberg group H n .

The Engulfing Property for Convex Functions in R R R n
Let us concentrate, first, on the one-dimensional case, i.e. n = 1. The following characterization holds (see Theorem 2 in [18], Theorem 5.1 in [14]): Theorem 2.1 Let u : R → R be a strictly convex and differentiable function. The following are equivalent: ii. there exists a constant K > 1 such that, if x, y ∈ R and s > 0 verify x ∈ S u (y, s), then y ∈ S u (x, K s); iii. there exists a constant K > 1 such that, for any x, y ∈ R, 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 : R → R be a convex function, with bounded sections, satisfying the engulfing property. Then, u is strictly convex and is in C 1 (R).
Given a strictly convex differentiable function u : R → R, one can consider the associated Monge-Ampère measure μ u defined on any Borel set A ⊂ R by where | · | denotes the Lebesgue measure. We say that the measure μ u has the (DC)doubling property if there exist constants α ∈ (0, 1) and C > 1 such that for every section S u (x, s) (here αS u (x, s) is the open convex set obtained by αcontraction 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 μ u is equivalent to the engulfing property for the function u; in particular, given u in E(R, K ), the constants α and C in (2.2) depend only on K . A Radon measure μ 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 R are called congruent if there exists an isometry of R that maps one of them onto the other. Since every open and bounded interval in R is a particular section for u, the (DC)-doubling property of μ u is trivially equivalent to the fact that μ u is a doubling measure. In particular, the constant A depends only on K . Now, noticing that μ 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 ) → u(x+r )−u(x)−u (x)r in our paper. More precisely in [14] (see Theorem 5.5) the authors prove the following: Condition (2.4) says that u is essentially symmetric around every point, and condition (2.5) says that it satisfies the so-called 2 condition at each point in R.
Hence, the behaviour of the measure μ u is related to the functions m u , M u : for every x ∈ R, r ∈ R + . These functions will be naturally extended to the ndimensional case and in H n , and will play a crucial role in the investigation of the engulfing for H -convex functions. For every fixed x ∈ R, denote by u x the function Then, M u (x, r ) ∈ {u x (±r )}, and M u (x, 2r ) ∈ {u x (±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 ∈ E(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 ≥ 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 ∈ E(R, K ) be a convex function with bounded sections. Then, with B 4 > 1 which depends only on K .
Proof Let us fix x ∈ R. The function u x defined in (2.7) is strictly convex and differentiable (see [11]), and belongs to E(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 (±2r ) ≥ 2 Let us now move to the case n ≥ 2. Given a differentiable function u : R n → R, as in the one-dimensional case (2.6), the functions m u , M u : R n × R + → R + are defined by for every x ∈ R n , r ∈ R + . Let us recall the following property, that will be critical when dealing with the engulfing in H n . Definition 2.1 (see Definition 2.1 in [21]) Let u : R n → R be a convex function. We say that u has round sections if there exists a constant τ ∈ (0, 1) with the following property: for every x ∈ R n , p ∈ ∂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 : R n → 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 ≥ 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 : R n → R satisfies one of the two equivalent conditions above, then u ∈ E(R n , K ), for a suitable K > 1 (see Theorem 3.9 in [21]). Let us finally notice that condition (2.12) is the n-dimensional version of condition (2.10): in the case n ≥ 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.

Convexity in the Heisenberg Group H H H n
The Heisenberg group 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 h of H n admits a stratification h = V 1 ⊕ V 2 with V 1 = span{X i , Y i ; 1 ≤ i ≤ n} being the first layer of the so-called horizontal vector fields, and V 2 = span{T } being the second layer which is one-dimensional. We assume [X i , Y i ] = −4T and the remaining commutators of basis vectors vanish. The exponential map exp : h → H n is defined in the usual way. By these commutator rules we obtain, using the Baker-Campbell-Hausdorff formula, that H n can be identified with R n × R n × R endowed with the non-commutative group law given by where x, y, x and y are in R n , t and t in R, and where · is the inner product in R n . Let us denote by e the neutral element in H n . Transporting the basis vectors of V 1 from the origin to an arbitrary point of the group by a left-translation, we obtain a system of left-invariant vector fields written as first order differential operators as follows For every positive λ, the non-isotropic Heisenberg dilation δ λ : H n → H n is defined by δ λ (x, y, t) = (λx, λy, satisfies the triangle inequality, thereby defining a distance on H n : this distance is the so-called Korányi-Cygan distance which is left-invariant and homogeneous, i.e. d g (δ λ (ξ ), δ λ (ξ )) = λd g (ξ, ξ ) for every λ > 0, ξ, ξ ∈ H n . We will set d g (e, ξ) = ξ g for every ξ ∈ H n . The Korányi-Cygan ball of center ξ 0 ∈ H n and radius r > 0 is given by The horizontal structure relies on the notion of horizontal plane. Given ξ 0 ∈ H n , the horizontal plane H ξ 0 associated to ξ 0 = (x 0 , y 0 , t 0 ) is the plane in H n defined by This is the plane spanned by the horizontal vector fields {X i , Y i } i at the point ξ 0 ; note that ξ ∈ H ξ if and only if ξ ∈ H ξ . A horizontal segment is a convex subset of a horizontal line, which is a line lying on a horizontal plane H ξ and passing though the Let ⊂ 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 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 so-called horizontal gradient, which is defined as the Here, k ( ) denotes the Folland-Stein space of functions having continuous derivatives up to order k with respect to the vector fields X i and ; the vector associated to D H ϕ with respect to the fixed scalar product is the horizontal gradient ∇ H ϕ(ξ ).
For general non-smooth functions ϕ : → R, the horizontal subdifferential ∂ H ϕ(ξ 0 ) of ϕ at ξ 0 ∈ is given by A central object of study within this paper is provided by the H -convex functions. First of all, we recall that a set ⊂ H n is said to be horizontally convex where is H -convex, there are several equivalent ways to define the concept of H -convexity for ϕ. The most intuitive one is to require the classical convexity of the function when restricted to any horizontal line within . The same definition can be rephrased by considering the group operation: the function ϕ : → R is said to be H -convex if, for every ξ 1 , ξ 2 ∈ with ξ 1 ∈ H ξ 2 and λ ∈ [0, 1], we have that (2.14) If the strict inequality holds in (2.14), for every ξ 1 = ξ 2 and λ ∈ (0, 1), then ϕ is said to be strictly H -convex. H -convex 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 ϕ :

H-Convex Functions with Round H-Sections and with Controlled H-Slope
As already seen in the Introduction, a horizontal notion of section was given in [13] for functions defined on a general Carnot group G. We will consider the particular case G = H n .
Let ϕ : H n → R be an H -convex function, and let us fix In this section we essentially introduce the notions of round H -sections (see Definition 3.1) and controlled H -slope (see Definition 3.2), proving their equivalence (see Theorem 3.1). Let us emphasize that these two properties for an H -convex 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 H -section, H -convexity and H -subdifferential.
In the following of the paper, for every function ϕ : H n → R, and for every ξ 0 ∈ H n , If ϕ is H -differentiable, then we will set ϕ ξ 0 ,∇ H ϕ(ξ 0 ) = ϕ ξ 0 . The following result holds: This contradicts Theorem 1.4 in [3].
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 H -convex function ϕ : H n → R has round H -sections if there exists a constant K 0 ∈ (0, 1) with the following property: for every ξ ∈ H n , p ∈ ∂ H ϕ(ξ ) and s > 0, there exists R > 0 such that In particular, (3.4) implies that every H -section of a function with round H -sections is a bounded set. Clearly, Definition 3.1 is the H n -version of Definition 2.1; let us stress that it relies upon the subriemannian structure of H n since, for every point ξ, we restrict our attention only to the horizontal plane H ξ .

Remark 3.1
Let ϕ : H n → R be H -convex, and consider the convex function ϕ ξ 0 ,v : is not a singleton. Indeed, suppose that p+λq ∈ ∂ H ϕ(ξ 0 ), for every λ ∈ [0, 1], with q = 0. Then, by taking v = q, we have that In the previous remark and in the following result, the H -convexity plays a fundamental role in order to obtain some regularity properties of the function involved.

5)
where the constant C depends only on K 0 in (3.4).
Proof First of all note that, for every ξ 0 ∈ H n and v ∈ V 1 \ {0}, the function ϕ ξ 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, ϕ is strictly H -convex. Let is not a singleton. This contradicts the fact that ϕ ξ 0 ,v (0) is differentiable. Finally, taking into account that the function ϕ ξ 0 ,v is convex, differentiable and with round sections with constant K 0 , for every ξ 0 ∈ H n and v ∈ V 1 , again, by Lemma 3.2 in [21], one has that there exists a constant C depending only on K 0 such that In the sequel, given an H -differentiable function ϕ : H n → R, we will deal with the functions m H ϕ , M H ϕ : H n × R + → R + that will take the place in H n of the functions m u and M u in R n . They are defined as follows: for every ξ ∈ H n , r > 0. A simple exercise shows that, if ϕ : H n → R is an H -differentiable and strictly H -convex function, then for every ξ ∈ H n , and r > 0, The next definition, inherited from the corresponding one in R n (see (2.12) (ξ 0 , s) . Pick a point ξ such that d g (ξ , ξ 0 ) = R; then, ξ ∈ ∂ S H ϕ (ξ 0 , s) and ξ = ξ 0 • exp v . From the H -convexity of ϕ ξ 0 on H n , we have that where K 1 is as in (3.7). Now, for every ξ ∈ H ξ 0 such that d g (ξ, ξ 0 ) = R K 1 , by (3.7) we have Hence, Suppose now that condition b. holds true. Proposition 3.2 entails that ϕ is Hdifferentiable. Consider K 0 as in (3.4), and fix ξ ∈ H n and r > 0: we have to prove (3.7), where K 1 is uniform, i.e. it does not depend on ξ and r . Set s = m H ϕ (ξ, r ) and define Since ϕ has round H -sections, R is not empty. Set R = min R; trivially, R = r , and The two relations above imply that Take α ∈ N such that K 0 > 2 −α , 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 −α in (3.7).
In the next result we investigate the properties of the function m H ϕ , in order to shed some light on a finer behaviour of the H -sections. Proof For every ξ ∈ H n , r > 0 and v ∈ V 1 , with v = 1, set The function m H ϕ is continuous, and strictly increasing w.r.t. r , since ϕ is strictly H -convex; thus, Hence, by the Berge Maximum Theorem (see, for instance, [2]) m H ϕ is continuous, and Let us show that the previous inequality is strict. The set {v ∈ V 1 : v = 1} is compact, and m H ϕ (ξ, ·, ·) is continuous, then there exist v and v such that m H ϕ (ξ, v, r ) = m H ϕ (ξ, r ) and m H ϕ (ξ, v , r ) = m H ϕ (ξ, r ). This implies that

Engulfing Property for H-Sections of H-Convex Functions
This section is devoted to the study of the engulfing property E (H , K ) for the Hsections of an H -convex 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 H -sections property, or, equivalently, the controlled H -slope (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 H -convex functions defined in H n .  iii. there exists a constant K > 1 such that, for any ξ ∈ H n , ξ ∈ H ξ , for any p ∈ ∂ H ϕ(ξ ) and q ∈ ∂ H ϕ(ξ ), Pr 1 (ξ )) .
In particular, if any of the conditions above holds, ϕ is H -differentiable.  [14], the function ϕ ξ,v satisfies the engulfing condition with constant 2K (K + 1). This is equivalent to say that From (4.1), we get that i.e., ϕ is in E(H , 2K (K + 1)).
In order to prove that ii. implies iii., let ξ = ξ • exp v and consider the convex function ϕ ξ,v . Note that p · v ∈ ∂ ϕ ξ,v (0) and q · v ∈ ∂ ϕ ξ,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 The second inequality in iii. gives Then, Let us recall that a set-valued map T : In particular, if ϕ is an H -convex function, then the H -subdifferential map ∂ H ϕ is an H -monotone set-valued map (see [10]). The property iii. above requires, in fact, a stronger control on the H -monotonicity, 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 H -sections property; the relationship between round Hsections, or, equivalently, controlled H -slope, and the engulfing property corresponds to the similar one in R n , for n ≥ 2:

Theorem 4.1 If ϕ : H n → R is an H -convex function with round H -sections, then ϕ satisfies the engulfing property E(H , K ), where K depends only on K 0 in (3.4).
Proof Since ϕ has round H -sections, Proposition 3.2 implies that ϕ is strictly Hconvex and H -differentiable. Let ξ ∈ S H ϕ (ξ, s) be such that ξ = ξ • exp(r v) for some v in V 1 , with v = 1 and r > 0; we will prove that ξ ∈ S H ϕ (ξ , K s) where K depends only on K 0 in (3.4).
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 R 2 ; we adapt his idea to the case of the first Heisenberg group H. Example 4.1 Consider the following differentiable and strictly convex function u : x 2 |y| 2/3 + 2|y| 4/3 |y| > |x| 3 . (4.9) The Monge-Ampère measure μ u (we recall that μ u is defined by μ u (E) = |∂u(E)| for every Borel set E ⊂ R 2 ) is absolutely continuous with respect to the Lebesgue measure | · |, and it verifies the condition μ ∞ , i.e. for any δ 1 ∈ (0, 1) there exists δ 2 ∈ (0, 1) such that: for every section S u (z, s), with z ∈ R 2 , and for every Borel set B ⊂ S u (z, s), (S u (z, s)) < δ 1 (see Definition 3.7 in [21]). This condition μ ∞ is stronger than the (DC)-doubling property (see, for example, relation (3.1.1) in [19]), i.e., there exist constants α ∈ (0, 1) and C > 1 such that for every z, s > 0 (here αS u (z, s) denotes the open convex set obtained by αcontraction of S u (z, τ ) with respect to its center of mass). In [20] and [17] it was shown that the (DC)-doubling property of the measure μ 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 D 2 u ; thus, u is not quasiuniformly convex (see Theorem 7.1-i. 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 ∇u(0, 0) = (0, 0), we have, for large r , m u ((0, 0), r ) = min Now let us consider the function ϕ : H → R defined by ϕ(x, y, t) = u(x, y), for all (x, y, t) ∈ H. This function ϕ is R 3 -convex, and hence H -convex. Since ϕ has not controlled H -slope, and hence has not round H -sections. However, since it is easy to see that ϕ enjoys the engulfing property.

H n H n H n -Sections of H-Convex Functions and Their Engulfing Properties
In this section we will present our new definition of section in H n . First of all, we will prove that these H n -sections are full-dimensional, i.e., they contain a Korányi-Cygan ball. This allows to construct a topology in H n , as we will see in the next Sect. 6. In the second part, we introduce the condition of engulfing E(H n , K ) for these new H nsections. It will not be a surprise that ϕ ∈ E(H n , K ) implies that ϕ ∈ 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 H -sections, or, equivalently, controlled H -slope. 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 H n -section: Definition 5.1 Let ϕ : H n → R be an H -convex function and let us fix ξ 0 ∈ H n . For a given s > 0, an H n -section of ϕ at height s, with p 0 ∈ ∂ H ϕ(ξ 0 ), is the set In case ϕ is H -differentiable at ξ 0 , we will denote the H n -section at ξ 0 with height s by S H n ϕ (ξ 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 (G, •) with homogeneous norm · G , there exists a constant C > 0 and an integer k ∈ N such that any ξ ∈ G can be expressed as ξ = ξ 1 • ξ 2 • . . . • ξ k , with ξ i ∈ exp(V 1 ) and ξ i G ≤ C ξ G , for every i. If G = H n , the mentioned k is exactly 3, for every n ≥ 1. In other words, every point ξ ∈ 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 ξ ∈ H n and r > 0, 2) Clearly, δ λ B(e, r ) = B(e, λr ), and the associated distance d in H n is left-invariant and homogeneous; hence, it is bi-Lipschitz equivalent to d g and to any other leftinvariant and homogeneous distance in H n . Moreover, due to the Folland-Stein Lemma, we have that, for every ξ ∈ H n and r > 0, where C is the constant in the mentioned lemma. Let us prove the first fundamental property of the H n -sections, namely S H n ϕ (ξ 0 , p 0 , s) is full-dimensional. Proposition 5.1 Let ϕ : H n → R be an H -convex function. Then, for every ξ 0 ∈ H n , p 0 ∈ ∂ H ϕ(ξ 0 ) and s > 0, there exists r > 0 such that Proof Without loss of generality, we set ξ 0 = e. Since the H -subdifferential map ∂ H ϕ brings compact sets into compact sets (see, for instance, Proposition 2.1 in [4]), there exists a positive constant R such that Moreover, since ϕ is locally Lipschitz (see Theorem 1.2 in [5]), there exists a positive constant L such that where C is the constant in the Folland-Stein Lemma. We will show that B g (e, r ) ⊂ S H n ϕ (e, p 0 , s). Take any ξ ∈ B g (e, r ); then, Note that ξ i ∈ B(e, r ) for i = 1, 2, 3. Then, for every p i ∈ ∂ H ϕ(ξ i ) (i = 1, 2), by (5.4) and (5.5) we have and (5.6) and (5.7) we get the claim.
Starting from these H n -sections, we introduce the following engulfing property: We will refer to (eng H n ) K in case we need to specify the constant K in the previous condition.
It is clear that for every ξ ∈ H n , r > 0. Hence the assertion holds.
In order to prove our main result concerning the engulfing property of the H nsections, an extension to the Heisenberg case of the inequalities (2.8), (2.9) and (2.11) turns out to be quite useful: where B 1 depends only on K . Hence we have max taking the maximum w.r.t. to v, with v = 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.
In the final part of this section we will prove our main result concerning the relationship between round H -sections and the engulfing property of the H n -sections. The proof will be quite technical, deserving a few previous estimates.
Let ϕ : H n → R be an H -convex function with round H -sections (with constant K 0 ). Then, ϕ ∈ E(H , K ), and has controlled H -slope (with constant K 1 ), where both K , K 1 depend on K 0 . Denote by γ any positive integer such that Thus, from (3.7), and by iterating inequality (5.12), we obtain Then, we have that where B 1 depends only on K 0 . Then, A result similar to Proposition 5.5, involving now the H n -sections S H n ϕ (ξ, s), holds, but the proof is much more delicate: Proposition 5.6 Let ϕ : H n → R be a function with round H -sections (with K 0 as in (3.4) A similar argument proves that Now, recalling that s = m H ϕ (ξ 0 , r ), the previous inequality gives Finally, (5.11) and (5.20) implies and the proof in finished.
In order to introduce and prove the main result of the section, we need the following The H -convexity of ϕ and the H -monotonicity of ∇ H ϕ give Again the H -convexity of ϕ and (5.23)-(5.25) give We are now in the position to prove the first part of our main result in Theorem 1.1:

Question 1
The assumption of the round H -sections property for an H -convex function ϕ is a sufficient condition in order to guarantee that ϕ satisfies the engulfing property E(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(H n , K ).

Question 2
In [13] the authors study the engulfing property for convex functions in a generic Carnot group G; as a matter of fact, in this more general framework, the related definition of G-sections (as in Definition 5.1) would be affected by the different geometry of the group 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 so-called horizontal line, i.e., a set {ξ • exp sv} s∈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 G-sections may have a very peculiar shape.

Question 3
By Theorem 3.3.10 in [19], the engulfing property for a convex function ϕ : R n → R implies the existence of C > 0 and p ≥ 1 such that for every 0 < r < s ≤ 1, x 0 ∈ R n , t > 0, and x ∈ S ϕ (x 0 , rt) we have the inclusion note that, under the assumptions above, the function ϕ is differentiable (see [11]). Under any suitable version of the engulfing property in H n , can a similar inclusion be proved in 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 ≥ 2, and let u : R n → R be a convex function. The following are equivalent: i. u is quasiuniformly convex function, i.e. u is not affine, u ∈ W 2,n loc and there exists a constant K ≥ 1 such that a.e. x ∈ R n ; (7.1) ii. u is differentiable and ∇u : R n → R n is quasiconformal, recalling that an injective map F : R n → R n is quasiconformal if F ∈ W 1,n loc and there exists a constant K ≥ 1 such that a.e. x ∈ R n ; (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 H n has been introduced and intensively studied (see for example [12]). In this paper we introduce the notion of H -controlled slope and round H -sections for an H -convex function but, at least to our knowledge, a horizontal notion of quasiuniform convexity for H -convex 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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