  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.
