## Journal of the Mathematical Society of Japan

### Needle decompositions and isoperimetric inequalities in Finsler geometry

Shin-ichi OHTA

#### Abstract

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

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

Dates
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.jmsj/1524038669

Digital Object Identifier
doi:10.2969/jmsj/07027604

Mathematical Reviews number (MathSciNet)
MR3787735

Zentralblatt MATH identifier
06902437

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1524038669

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