The anisotropic total variation and surface area measures
type: Item
instance of: scholarly article
title: The anisotropic total variation and surface area measures
DOI: 10.1007/978-3-031-26300-2_11
author: Liran Rotem
publication date: 2024-09-20T00:00:00Z
review text: 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 \ma…
reviewed by: Julià Cufí
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 7916425
zbMATH Keywords: surface area measures
zbMATH Keywords: semi-continuous log-concave functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7916425