Needle decompositions and isoperimetric inequalities in Finsler geometry

Shin-ichi OHTA
2018 Journal of the Mathematical Society of Japan  
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Lévy-Gromov, Bakry-Ledoux and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (socalled 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 nonbranching
more » ... tially nonbranching metric measure spaces satisfying the curvature-dimension condition, in particular including 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 ) (in the sense of Lott-Sturm-Villani) for N = 0 is also included, it would be of independent interest. Preliminaries We review basic facts in Finsler geometry and optimal transport theory necessary in our discussion.
doi:10.2969/jmsj/07027604 fatcat:qafkzxcuyrhrxlqgehtcjznuia