# A disintegration technique for locally affine partitions of $\mathbb{R}^d$ and related divergence formulas

### (SISSA Trieste)

**Event:** ERC School on Analysis in Metric Spaces and Geometric Measure Theory

**Date:** Jan 13, 2011,
**time:** 15:10

**Place:** Centro De Giorgi, Scuola Normale Superiore

**Abstract.** We present a technique which permits to show, in several

problems of variational nature, that each conditional probability

obtained by the disintegration of the Lebesgue measure on certain

Borel partitions of $\mathbb{R}^d$ into convex sets of linear

dimension $k=0,...,d$ is equivalent to the $k$-dimensional Hausdorff

measure of the set on which it is concentrated.

The problem lies in the fact that we consider partitions which are a

priori just Borel, so that we cannot use Area or Coarea Formulas;

moreover, in dimension $d$ greater or equal than 3, there are Borel

partitions in segments for which the conditional probabilities of the

Lebesgue measure are Dirac deltas.

As a byproduct of this technique, the vector fields giving at each

point of $\mathbb{R}^d$ the directions of the linear span of the

convex set through that point satisfy a local divergence formula on

the sets of a countable covering of $\mathbb{R}^d$.

Two applications are given by a regularity result for convex functions

(joint work with L. Caravenna) and a characterization of optimal

transport plans for the Monge-Kantorovich problem w.r.t. a convex norm

in $\mathbb{R}^d$ (joint work with S. Bianchini).