CRM: Centro De Giorgi
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ERC School on Analysis in Metric Spaces and Geometric Measure Theory

A Disintegration Technique for Locally Affine Partitions of Rd 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 Rd 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 Rd the linear span of the convex set through that point satisfy a local divergence formula on the sets of a countable covering of Rd. 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 Rd (joint work with S. Bianchini).

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