Needle decompositions and isoperimetric inequalities in Finsler geometry

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on needle decompositions associated with 1-Lipschitz functions, inspired by convex geometry and optimal transport theory. Cavalletti and Mondino subsequently generalized Klartag's technique to essentially non-branching 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.