Calculus of Variations and Geometric Measure Theory

L. Caravenna - S. Daneri

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

created by daneri on 27 Apr 2009
modified on 29 Sep 2010


J. Funct. Anal.

Inserted: 27 apr 2009
Last Updated: 29 sep 2010

Year: 2010


We consider the disintegration of the Lebesgue measure on the graph of a convex function $f:\R^{n}\rightarrow \R$ 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.

Keywords: Hausdorff dimension, disintegration of measures, faces of a convex function, conditional measures, absolute continuity, Divergence Formula