Abstract
In this paper we investigate the property of engulfing for H-convex 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 H-convex 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 quasi-distance 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 H-sections for an H-convex function, and by its connection with the engulfing properties.
Highlights
Given a convex function u : Rn → R, for every x0 ∈ Rn, p ∈ ∂u(x0), and s > 0, we will denote by Su(x0, p, s) the section of u at x0 with height s, defined as followsSu(x0, p, s) = x ∈ Rn : u(x) − u(x0) − p · (x − x0) < s ; (1.1)The Engulfing Property for Sections of Convex Functions...in case u is differentiable at x0, we will denote the section by Su(x0, s), for short
In this paper we focus on horizontally convex functions φ (H -convex functions) on the Heisenberg group Hn, that is the simplest Carnot group of step 2
We prove that every H -convex function with round H -sections satisfies the engulfing property E(Hn, K ) in Definition 1.1
Summary
Given a convex function u : Rn → R, for every x0 ∈ Rn, p ∈ ∂u(x0), and s > 0, we will denote by Su(x0, p, s) the section of u at x0 with height s, defined as follows. ), and the family of sections of u consists of the usual balls in Rn. 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. Our idea takes inspiration from the notion of H -section in (1.4), together with the property that any pair of points in Hn can be joined by at most three consecutive horizontal segments These facts lead us to define full-dimensional sections that arise as a sort of composition in three steps of “thin” H -sections.
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