Journal of the Mathematical Society of Japan

Needle decompositions and isoperimetric inequalities in Finsler geometry

Shin-ichi OHTA

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Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Lévy–Gromov, Bakry–Ledoux, Bayle and 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 $\mathrm{CD}(K,N)$ for $N=0$ is also included, it would be of independent interest.

Article information

J. Math. Soc. Japan, Volume 70, Number 2 (2018), 651-693.

First available in Project Euclid: 18 April 2018

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Zentralblatt MATH identifier

Primary: 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20]
Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting

Finsler geometry Ricci curvature localization isoperimetric inequality


OHTA, Shin-ichi. Needle decompositions and isoperimetric inequalities in Finsler geometry. J. Math. Soc. Japan 70 (2018), no. 2, 651--693. doi:10.2969/jmsj/07027604.

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