ERC School on Analysis in Metric Spaces and Geometric Measure Theory

A Disintegration Technique for Locally Affine Partitions of R^d and Related Divergence Formulas

speaker: Sara Daneri (University of Leipzig)

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 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 the directions of R^d the linear span of the convex set through that point satisfy a local divergence formula on the sets of a countable covering of 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 R^d (joint work with S. Bianchini).

timetable:
Thu 13 Jan, 15:10 - 15:40, Aula Dini
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