We consider the isodiametric problem in Carnot groups, i.e., the problem of finding sets, called isodiametric, that maximize the Haar measure among all sets with a fixed diameter. In the Euclidean setting, it is well known that isodiametric sets are the balls. For non abelian Carnot groups, the situation is significantly different. For instance one can always find a homogeneous distance for which balls are not isodiametric. This fact has various consequences, related in particular to rectifiability and density properties of the space. We will also give more refined results in the case of the Heisenberg group.
Copyright. All Rights Reserved.