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

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 is an element of partial derivative C and Q is an element of int(C), both belonging to a horizontal line l, then an open truncated subRiemannian cone around l 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 H x R of the Heisenberg group with the real line have hyperplanes as boundaries.