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

TL;DR

This paper establishes a cone property for horizontally convex sets in step two Carnot groups and uses it to classify monotone sets, showing they have hyperplanes as boundaries in certain groups.

Contribution

It proves a new cone property for convex sets in step two Carnot groups and applies it to classify monotone sets, revealing their boundary structure.

Findings

Horizontally convex sets satisfy a cone property in step two Carnot groups.

Monotone sets in \\mathbb{H} \\times \\mathbb{R} have hyperplanes as boundaries.

Abstract

In the setting of step two 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 $ 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.