Leaves decompositions and optimal transport of vector measures

Krzysztof J. Ciosmak

Published 2019-05-06Version 1

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 this sets. We consider a disintegration of the Lebesgue measure with respect to this partition. We prove that almost every conditional measure associated to a set of dimension $m$ is equivalent to the restriction of the $m$-dimensional Hausdorff measure to its support. We provide a conterexample to the conjecture of Klartag that, given a vector measure on $\mathbb{R}^n$ with total mass zero, the conditional measures, with respect to 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.