Needle decompositions and isoperimetric inequalities in Finsler geometry

By Shin-ichi Ohta. 2015

47 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/a07999c1cf31fe6e0b62a7daa2417d13b2884f3d

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

The globalization theorem for the Curvature-Dimension condition (2016, background): “This allowed Cavalletti and Mondino (see [27, 28]) to obtain a series of sharp geometric and analytic inequalities for CDloc(K,N) spaces as above, in particular extending from the smooth Riemannian setting the sharp Lévy-Gromov [42] and Milman [54] isoperimetric inequalities, as well as the sharp B”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, background): “Another extension of [13, 45] is given by Ohta [50, 51] to the setting of (not necessarily reversible) Finsler manifolds.”