Leaves decompositions in Euclidean spaces

TL;DR

This paper extends the localisation technique to multiple constraints in Euclidean spaces, establishing a partition into convex sets where a Lipschitz map acts as an isometry, and analyzing the log-concavity of measures on these sets.

Contribution

It introduces a new partitioning method for Euclidean spaces under multiple constraints, proving the existence of convex sets where Lipschitz maps are isometries and analyzing measure disintegration.

Findings

Partition of Euclidean space into convex sets where the map is an isometry

Conditional measures on these sets are log-concave for almost every set

Extension of results to weighted Riemannian manifolds under curvature-dimension conditions

Abstract

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, 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.