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

@article{Morbidelli2018OnTI,
title={On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets},
author={D. Morbidelli},
journal={arXiv: Metric Geometry},
year={2018}
}
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… Expand
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