The anisotropic total variation and surface area measures (Q6608552)

The goal of this paper is to prove a formula for the first variation of the integral of a log-concave function that justifies a definition of the surface area measure of such a function. To formulate it some notions and notations are needed. If \(K\) denotes a convex body in \(\mathbb{R}^n\), \(\eta_K\) will be the Gauss map from the boundary of \(K\) to the unit sphere \(\mathbb{S}^{n-1}\). One will consider log-concave functions as a generalization of convex bodies. If \(f, g\) are two log-concave functions their sum is given by the sup-convolution \N\[\N(f *g) (x) = \sup_{y \in \mathbb{R}^n} (f(y) g(x-y))\N\] \Nand for two upper semi-continuous log-concave functions \(f, g\) the first variation of the integral of \(f\) in the direction of \(g\) is given by \N\[\N\delta(f , g) = \lim _{t\to 0} \frac{\int f * (t\cdot g) - \int f}{t}, \N\]\Nwhere \(\int f\) is the Lebesgue integral. If \(g\) is a log-concave function, \(h_g: \mathbb{R}^n \to \mathbb{R}\) is the support function of \(g\), defined by \(h_g = (- \log g)^*\), where \(\alpha^* (x) = \sup_{y \in \mathbb{R}^n} (\langle x , y\rangle - \alpha(y))\) is the Legendre transform. \N\NNow one can define the two measures needed to express the first variation of the integral of a function. Fix a log-concave function \(f : \mathbb{R}^n \to \mathbb{R}\) with \(0 < \int f < \infty\), and write \(f = e ^{-\phi}\) for a convex function \(\phi : \mathbb{R}^n \to (- \infty , \infty]\). Then \N\begin{itemize}\N\item[1)] The measure \(\mu_f\) is a measure on \(\mathbb{R}^n\) defined as the push-forward of the measure \(f dx\) by \(\operatorname{grad}(\phi)\). \N\item[2)] The measure \(\nu_f\) is a measure on the sphere \(\mathbb{S}^{n-1}\), defined as the push-forward of the measure \(f\cdot H^{n-1}\) restricted to the boundary of \(K_f\) by the Gauss map \(\eta_{K_f}\) of the boundary of \(K_f\) in \(\mathbb{S}^{n-1}\). \N\end{itemize}\NHere \(H^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure and \(K_f = \{ x \in \mathbb{R}^n : f(x) > 0 \}\). The pair \((\mu_f, \nu_f)\) is the surface area measures for a log-concave function \(f\) with \(0 < \int f < \infty\). \N\NThe main result proved here is the following one: \NFix \(f, g\) two upper semi-continuous log-concave functions with \(0 < \int f < \infty\). Then\N\[\N\delta(f , g) = \int_{\mathbb{R}^n} h_g d\mu_f + \int_{\mathbb{S}^{n-1}} h_{K_g} d\nu_f. \N\]\NThis result was previously been proved but assuming some regularity conditions over the functions \(f\) and \(g\).\N\NFor the entire collection see [Zbl 1527.46003].