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The disintegration of the Lebesgue measure on the faces of a convex function

TitleThe disintegration of the Lebesgue measure on the faces of a convex function
Publication TypeJournal Article
Year of Publication2010
AuthorsCaravenna, L, Daneri, S
JournalJ. Funct. Anal. 258 (2010) 3604-3661
Abstract

We consider the disintegration of the Lebesgue measure on the graph of a convex function f:\\\\Rn-> \\\\R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure of the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets.

URLhttp://hdl.handle.net/1963/3622
DOI10.1016/j.jfa.2010.01.024

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