Annales AcademiŠ Scientiarum FennicŠ
Volumen 36, 2011, 245-260
UniversitÚ de Nice Sophia Antipolis, UMR CNRS 6621, Laboratoire J.A. DieudonnÚ
Parc Valrose, 06108 Nice Cedex 02, France; rigot 'at' unice.fr
Abstract. The classical isodiametric inequality in the Euclidean space says that balls maximize the volume among all sets with a given diameter. We consider in this paper the case of Carnot groups. We prove that for any non abelian Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-CarathÚodory distances and prove that this also fails for these distances as soon as there are length minimizing curves that stop to be minimizing in finite time. Next we study some connections with the comparison between Hausdorff and spherical Hausdorff measures, rectifiability and the generalized 1/2-Besicovitch conjecture, giving in particular some cases where this conjecture fails.
2000 Mathematics Subject Classification: Primary 28A75; Secondary 53C17.
Key words: Isodiametric inequality, homogeneous groups, densities.
Reference to this article: S. Rigot: Isodiametric inequality in Carnot groups. Ann. Acad. Sci. Fenn. Math. 36 (2011), 245-260.
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