Leaves decompositions in Euclidean spaces and optimal transport of vector measures

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.