We consider the disintegration of the Lebesgue measure on the graph of a convex function 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 on 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.

The Disintegration of the Lebesgue Measure on the Faces of a Convex Function

DANERI S
2010

Abstract

We consider the disintegration of the Lebesgue measure on the graph of a convex function 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 on 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/7386
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