Isodiametric inequality in Carnot groups

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 Carnot group equipped with a Haar measure one can find a homogeneous distance for which this fails to hold. We also consider Carnot-Caratheodory 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.Comment: 14 page

Quasiminimal crystals with a volume constraint and uniform rectifiability

AbstractWe establish here, in a quite general context, uniform rectifiability properties for quasiminimal crystals with a volume constraint. Namely we prove that to any quasiminimal crystal with a volume constraint corresponds a unique equivalent open set whose boundary is Ahlfors-regular and which satisfies the so-called condition B. Moreover implicit bounds in these properties, which imply the uniform rectifiability of the boundary, can be chosen universal. As a consequence we give a universal upper bound for the number of connected components of reduced quasiminimizers and we also prove that quasiminimal crystals with a volume constraint actually satisfy, in some universal way, an apparently stronger quasiminimality condition where admissible perturbations are not required to be volume-preserving anymore