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” forhorizontally convex sets. Namely, we prove that, given a horizontally convex set C,a pair of points P ¬ C and Q ¬ int(C), both belonging to a horizontal line , thenan open truncated subRiemannian cone around and with vertex at P is containedin C.We apply our result to the problem of classification of horizontally monotone setsin Carnot groups. We are able to show that monotone sets in the direct product H×Rof the Heisenberg group with the real line have hyperplanes as boundaries.