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

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

Isoperimetric sets on Carnot groups

We prove the existence of isoperimetric sets in any Carnot group,that is, sets minimizing the intrinsic perimeter among all measurable setswith prescribed Lebesgue measure. We also show that, up to a null set,these isoperimetric sets are open, bounded, their boundary is Ahlfors-regularand they satisfy the condition B. Furthermore, in the particular case of theHeisenberg group, we prove that any reduced isoperimetric set is a domain ofisoperimetry. All these properties are satisfied with implicit constants thatdepend only on the dimension of the group and on the prescribed Lebesguemeasure