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2020 On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets
Daniele Morbidelli
Publ. Mat. 64(2): 391-421 (2020). DOI: 10.5565/PUBLMAT6422002

Abstract

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.

Citation

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Daniele Morbidelli. "On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets." Publ. Mat. 64 (2) 391 - 421, 2020. https://doi.org/10.5565/PUBLMAT6422002

Information

Received: 3 September 2018; Revised: 30 January 2019; Published: 2020
First available in Project Euclid: 3 July 2020

zbMATH: 07236048
MathSciNet: MR4119259
Digital Object Identifier: 10.5565/PUBLMAT6422002

Subjects:
Primary: 53C17
Secondary: 52A01

Keywords: Carnot groups , monotone sets , subRiemannian distance

Rights: Copyright © 2020 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.64 • No. 2 • 2020
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