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.

1 aCaravenna, Laura1 aDaneri, Sara uhttp://hdl.handle.net/1963/362200354nas a2200097 4500008004100000245007900041210006900120260001000189100002100199856003600220 2009 en d00aThe Disintegration Theorem and Applications to Optimal Mass Transportation0 aDisintegration Theorem and Applications to Optimal Mass Transpor bSISSA1 aCaravenna, Laura uhttp://hdl.handle.net/1963/5900