**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).

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

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