The engulfing property for sections of convex functions in the Heisenberg group and the associated quasi--metric

TL;DR

This paper introduces a new notion of sections called ${ mf H}^n$-sections for $H$-convex functions on the Heisenberg group, establishing an engulfing property that leads to a pseudo-metric similar to Euclidean cases.

Contribution

It defines ${ mf H}^n$-sections and an associated engulfing property for $H$-convex functions on the Heisenberg group, extending Euclidean concepts to a sub-Riemannian setting.

Findings

${ mf H}^n$-sections are larger unions of horizontal sections.

Engulfing property leads to a pseudo-metric in ${ mf H}^n$.

Round $H$-sections are key to establishing engulfing properties.

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, 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 pseudo-metric in ${\mathbb{H}}^n$ in a way similar to Aimar, Forzani and Toledano in the Euclidean case. 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.