Existence of isoperimetric regions in sub-Finsler nilpotent groups

Abstract

We consider a nilpotent Lie group with a bracket-generating distribution $\hhh$ and an asymmetric left-invariant norm $\|\cdot\|_K$ induced by a convex body $K\subseteq\hhh$ containing $0$ in its interior. In this paper we prove the existence of minimizers of the perimeter functional $P_K$ associated to $\|\cdot\|_K$ under a volume (Haar measure) constraint. Our result generalizes the one of Leonardi and Rigot for Carnot groups.