The anisotropic total variation and surface area measures

Abstract

We prove a a formula for the first variation of the integral of a log-concave function, which allows us to define the surface area measure of such a function. The formula holds in complete generality with no regularity assumptions, and is intimately related to the notion of anisotropic total variation and to anisotropic coarea formulas. This improves previous partial results by Colesanti and Fragal\`a, by Cordero-Erausquin and Klartag and by the author.