Leaves decompositions in Euclidean spaces

Abstract

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u:{\mathbb{R}}^n\to {\mathbb{R}}^m$, $m\le n$, we define and prove the existence of a partition of ${\mathbb{R}}^n$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.

accepted in Journal de Mathématiques Pures et Appliquées; the present preprint is formed from arXiv:1905.02182, which has been split; 28 pages