In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter.In particular, given an isodiametric set, and up to negligible sets, we prove that its boundary is given by the graphs of two locally Lipschitz functions.Moreover, in the restricted class of rotationally invariant sets, we give a quite complete characterization of any compact (rotationally invariant) isodiametric set. More specifically, its Steiner symmetrization with respect to the Cn-planeis shown to coincide with the Euclidean convex hull of a CC-ball. At the same time, we also prove quite unexpected non-uniqueness results.
Isodiametric sets in the Heisenberg group / Leonardi, Gian Paolo; Rigot, Severine; Vittone, Davide. - In: REVISTA MATEMATICA IBEROAMERICANA. - ISSN 0213-2230. - STAMPA. - 2012, 28:28 (4)(2012), pp. 999-1024. [10.4171/rmi/700]
Isodiametric sets in the Heisenberg group
LEONARDI, Gian Paolo;
2012-01-01
Abstract
In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter.In particular, given an isodiametric set, and up to negligible sets, we prove that its boundary is given by the graphs of two locally Lipschitz functions.Moreover, in the restricted class of rotationally invariant sets, we give a quite complete characterization of any compact (rotationally invariant) isodiametric set. More specifically, its Steiner symmetrization with respect to the Cn-planeis shown to coincide with the Euclidean convex hull of a CC-ball. At the same time, we also prove quite unexpected non-uniqueness results.File | Dimensione | Formato | |
---|---|---|---|
RigLeoVit_RMI2012.pdf
accesso aperto
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
415.69 kB
Formato
Adobe PDF
|
415.69 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione