00653nas a2200109   4500008004100000245009700041210006900138260001000207520027300217100001700490856003600507       2011                        en  d00aDimensional Reduction and Approximation of Measures and Weakly Differentiable Homeomorphisms0 aDimensional Reduction and Approximation of Measures and Weakly D  bSISSA3 aThis thesis is devoted to the study of two different problems: the properties of the disintegration of the Lebesgue measure on the faces of a convex function and the existence of smooth approximations of bi-Lipschitz orientation-preserving homeomorphisms in the plane.1 aDaneri, Sara  uhttp://hdl.handle.net/1963/534800984nas a2200109   4500008004300000245008100043210006900124520060700193100002100800700001700821856003600838       2010                        en_Ud  00aThe disintegration of the Lebesgue measure on the faces of a convex function0 adisintegration of the Lebesgue measure on the faces of a convex 3 a<p>We consider the disintegration of the Lebesgue measure on the graph of a convex function f:\\\\Rn-&gt; \\\\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.</p>1 aCaravenna, Laura1 aDaneri, Sara  uhttp://hdl.handle.net/1963/3622