On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets

Abstract

In the setting of two-step Carnot groups we show a “cone property” for horizontally convex sets. Namely, we prove that, given a horizontally convex set $C$, a pair of points $P\in \partial C$ and $Q\in \operatorname{int} (C)$, both belonging to a horizontal line $\ell$, then an open truncated subRiemannian cone around $\ell$ and with vertex at $P$ is contained in $C$.

We apply our result to the problem of classification of horizontally monotone sets in Carnot groups. We are able to show that monotone sets in the direct product $\mathbb{H}\times\mathbb{R}$ of the Heisenberg group with the real line have hyperplanes as boundaries.