  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.
