Klartag recently gave a beautiful alternative proof of the isoperimetricinequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weightedRiemannian manifolds. Klartag's approach is based on a generalization of thelocalization method (so-called needle decompositions) in convex geometry,inspired also by optimal transport theory. Cavalletti and Mondino subsequentlygeneralized the localization method, in a different way more directly alongoptimal transport theory, to essentially non-branching metric measure spacessatisfying the curvature-dimension condition. This class in particular includesreversible (absolutely homogeneous) Finsler manifolds. In this paper, weconstruct needle decompositions of non-reversible (only positively homogeneous)Finsler manifolds, and show an isoperimetric inequality under boundedreversibility constants. A discussion on the curvature-dimension conditionCD$(K,N)$ for $N = 0$ is also included, it would be of independent interest.
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机译：最近，克拉拉塔格（Klartag）在加权黎曼流形上给出了Levy-Gromov，Bakry-Ledoux，Bayle和E.Milman的等距不等式的漂亮替代证明。 Klartag的方法基于凸几何中的局部化方法（所谓的针头分解）的一般化，也受到最佳传输理论的启发。 Cavalletti和Mondino随后以最优运输理论的另一种更直接的方式将定位方法推广到了满足曲率维条件的本质上无分支的度量度量空间。此类尤其包括可逆（绝对均匀）的Finsler流形。在本文中，我们构造了不可逆（仅是正均匀的）芬斯勒流形的针头分解，并在有界可逆常数下显示了等长不等式。对于$ N = 0 $的曲率维数条件CD $（K，N）$的讨论也包括在内，这将引起人们的关注。
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