# Leaves decompositions in Euclidean spaces and optimal transport of vector measures

## TL;DR

This paper studies partitions of Euclidean spaces induced by 1-Lipschitz maps, explores log-concavity of disintegrated measures, and develops a theory of optimal transport for vector measures, partially confirming and refuting conjectures by Klartag.

## Contribution

It introduces a novel partitioning approach based on 1-Lipschitz maps, proves log-concavity of conditional measures, and develops optimal transport theory for vector measures, addressing conjectures in the field.

## Key findings

- Conditional measures are log-concave for almost every partition set.
- Counterexample provided for a conjecture on total mass zero of conditional measures.
- Optimal transport theory for vector measures developed and applied.

## Abstract

For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ 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 $CD(\kappa,N)$ for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag. We provide a counterexample to another conjecture of Klartag that, given a vector measure on $\mathbb{R}^n$ with total mass zero, the conditional measures, with respect to partition obtained from a certain $1$-Lipschitz map, also have total mass zero. We develop a theory of optimal transport for vector measures and use it to answer the conjecture in the affirmative provided a certain condition is satisfied.