Hessian measures of semi-convex functions and applications to support measures of convex bodies
type: Item
instance of: scholarly article
title: Hessian measures of semi-convex functions and applications to support measures of convex bodies
zbMATH Open document ID: 0973.52003
author: A. Colesanti
author: Daniel Hug
published in: Manuscripta Mathematica
publication date: 2001-09-06T00:00:00Z
review text: The paper contains deep results on support measures (thus, in particular, on curvature and surface area measures) of convex bodies and, more generally, of sets with positive reach. \textit{R. Schneider} proved [see Theorem 4.6.5 in his book ``Convex bodies: The Brunn-Minkowski theory'' (1993; Zbl 0798.52001)] that the \(j\)th curvature measure \(C_j(K,\cdot)\) (the \(j\)th surface area measure \(S_j(K,\cdot)\)) of a convex body \(K\) is absolutely continuous with resp. to the Hausdorff measure of a suitable dimension. The authors obtain this result as a consequence of Theorem 0.1, which states that for every convex body \(K \subset \text{R}^d\) and \(j \in \{0,\dots ,d-1\}\), if \(\alpha \subset \text{bd} K\) is a Borel set with \(\sigma\)-finite \(j\)-dimensional Hausdorff measure, then \[ {d-1 \choose j} C_j(K,\alpha)= \int_\alpha H^{d-1-j}(N(K,x)\cap S^{d-1})d{\mathcal H}^j(x), \] where \(N(K,x)\) is the normal cone of \(K\) at \(x\). Furthermore, a similar assertion holds for surface area measures. They study Hessian measures \(F_k\) of semi-convex functions (the notion introduced in their earlier paper). These measures are analogues of the support measures for sets with positive reach: they are defined as coefficients in a local Steiner type formula ((4) p. 14) and satisfy a Crofton type formula (Theorem 2.1). Moreover, the measure \(F_k(u,\cdot)\) of a semi-continuous function \(u\) is absolutely continuous with resp. to the Hausdorff measure \({\mathcal H}^k\) (Theorem 3.1). Theorem 5.4 describes a simple connection between the support measures of a set \(X\) with positive reach and the Hessian measures of its distance function \(d_X\). Similarly, Corollary 5.9 describes a connection between the support measures of a convex body \(K\) and the Hessian measures of its support function \(h_K\). Using these theorems (Theorem 5.4 and Corollary 5.9) the authors deduce results on support measures of a set \(X\) with positive reach from corresponding results on the H…
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1423147
zbMATH Keywords: convex body
zbMATH Keywords: set with positive reach
zbMATH Keywords: convex function
zbMATH Keywords: semi-convex function
zbMATH Keywords: Hessian measure
zbMATH Keywords: support measure
zbMATH Keywords: curvature measure
zbMATH Keywords: surface area measure
reviewed by: Maria Moszyńska
MaRDI profile type: Publication
full work available at URL: s002290050015
OpenAlex ID: W2056784142
DOI: 10.1007/S002290050015
description: scientific article; zbMATH DE number 1423147