Existence of isoperimetric regions in sub-Finsler nilpotent groups

Abstract We consider a nilpotent Lie group with a bracket-generating distribution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℋ</m:mi> </m:math> {\mathcal{ {\mathcal H} }} and an asymmetric left-invariant norm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>∣</m:mo> <m:mo>⋅</m:mo> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>K</m:mi> </m:mrow> </m:msub> </m:math> {| \cdot | }_{K} induced by a convex body <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> <m:mo>⊆</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msup> </m:math> K\subseteq {{\mathbb{R}}}^{k} containing 0 in its interior. In this study, we prove the existence of minimizers of the perimeter functional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>K</m:mi> </m:mrow> </m:msub> </m:math> {P}_{K} associated with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>∣</m:mo> <m:mo>⋅</m:mo> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>K</m:mi> </m:mrow> </m:msub> </m:math> {| \cdot | }_{K} under a volume (Haar measure) constraint.