Needle decompositions and isoperimetric inequalities in Finsler geometry

TL;DR

This paper extends needle decomposition techniques to non-reversible Finsler manifolds, establishing isoperimetric inequalities under certain conditions and discussing curvature-dimension properties, thus broadening geometric analysis tools.

Contribution

It constructs needle decompositions for non-reversible Finsler manifolds and proves related isoperimetric inequalities, expanding the scope of localization methods in geometric analysis.

Findings

Needle decompositions are possible in non-reversible Finsler manifolds.

An isoperimetric inequality is established under bounded reversibility constants.

Discussion on curvature-dimension condition CD(K,N) for N=0 is included.

Abstract

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvature-dimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric inequality under bounded reversibility constants. A discussion on the curvature-dimension condition CD$(K,N)$ for $N = 0$ is also included, it would be of independent interest.