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

2018/08/20 by Morbidelli, Daniele · 1 citation
#52A01 #53C17 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1808.06513

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∈ ∂ C and Q∈ 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 ×ℝ of the Heisenberg group with the real line have hyperplanes as boundaries.

Cited by

• A note on pliability and the openness of the multiexponential map in Carnot groups

Related

• Precisely monotone sets in step-2 rank-3 Carnot algebras
• A rectifiability result for finite-perimeter sets in Carnot groups
• A note on pliability and the openness of the multiexponential map in Carnot groups
• Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter
• Geometric properties of extremal sets for the dimension comparison in Carnot groups of step 2