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. 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-section, as well as a new condition of engulfing associated to the Hn-sections, for an H-convex function defined in Hn. These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the Hn-sections, with their engulfing property, will lead to the definition of a quasi-distance in Hn 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.