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 is an element of partial derivative C and Q is an element of 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 x R of the Heisenberg group with the real line have hyperplanes as boundaries.

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

Morbidelli, Daniele
2020

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 is an element of partial derivative C and Q is an element of 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 x R of the Heisenberg group with the real line have hyperplanes as boundaries.
2020
Morbidelli, Daniele
File in questo prodotto:
File Dimensione Formato  
Inner_cone.pdf

accesso riservato

Tipo: Versione (PDF) editoriale
Licenza: Licenza per accesso riservato
Dimensione 438.12 kB
Formato Adobe PDF
438.12 kB Adobe PDF   Visualizza/Apri   Contatta l'autore

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/771849
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact