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arXiv · 1808.06513

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

Daniele Morbidelli
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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.

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Daniele Morbidelli. 2018-08-20. On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets. https://arxiv.org/abs/1808.06513

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