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

Sara Daneri

(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).