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.

}, doi = {10.1016/j.jfa.2010.01.024}, url = {http://hdl.handle.net/1963/3622}, author = {Laura Caravenna and Sara Daneri} } @mastersthesis {2009, title = {The Disintegration Theorem and Applications to Optimal Mass Transportation}, year = {2009}, school = {SISSA}, url = {http://hdl.handle.net/1963/5900}, author = {Laura Caravenna} }