On the ${\mathbb H}$-cone-functions for H-convex sets

TL;DR

This paper investigates the existence and properties of H-cone-functions for H-convex sets in the Heisenberg group, revealing conditions under which such functions exist and exploring their geometric implications.

Contribution

The paper introduces a new approach using an extension of Fenchel's convex family concept to characterize when H-cone-functions exist for H-convex sets in the Heisenberg group.

Findings

Derived precise conditions for the existence of H-cone-functions.

Identified shape constraints on H-convex sets for cone-function existence.

Provided multiple examples illustrating these conditions.

Abstract

Given a compact and H-convex subset $K$ of the Heisenberg group ${\mathbb H}$, with the origin $e$ in its interior, we are interested in finding a homogeneous H-convex function $f$ such that $f(e)=0$ and $f\bigl|_{\partial K}=1$; we will call this function $f$ the ${\mathbb H}$-cone-function of vertex $e$ and base $\partial K$. While the equivalent version of this problem in the Euclidean framework has an easy solution, in our context this investigation turns out to be quite entangled, and the problem can be unsolvable. The approach we follow makes use of an extension of the notion of convex family introduced by Fenchel. We provide the precise, even if awkward, condition required to $K$ so that $\partial K$ is the base of an ${\mathbb H}$-cone-function of vertex $e.$ Via a suitable employment of this condition, we prove two interesting binding constraints on the shape of the set $K,$ together with several examples.