The anisotropic total variation and surface area measures

TL;DR

This paper establishes a general formula for the first variation of the integral of log-concave functions, enabling the definition of surface area measures without regularity assumptions, advancing the understanding of anisotropic total variation.

Contribution

It introduces a comprehensive formula for the first variation of log-concave functions' integrals, linking it to anisotropic total variation and surface area measures, improving prior partial results.

Findings

General formula for first variation of log-concave functions

Definition of surface area measure without regularity assumptions

Enhanced connection to anisotropic total variation and coarea formulas

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.