## On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets.(English)Zbl 07236048

Summary: 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 $$P \in \partial C$$ and $$Q \in \operatorname{int} (C)$$, 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.

### MSC:

 53C17 Sub-Riemannian geometry 52A01 Axiomatic and generalized convexity

### Keywords:

subriemannian distance; Carnot groups; monotone sets
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### References:

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