Absolute continuity for curvature measures of convex sets. II
type: Item
instance of: scholarly article
title: Absolute continuity for curvature measures of convex sets. II
zbMATH Open document ID: 0954.52006
DOI: 10.1007/PL00004765
author: Daniel Hug
published in: Mathematische Zeitschrift
publication date: 2001-02-14T00:00:00Z
review text: In continuation of his previous work [Math. Nachr. 195, 139-158 (1998; Zbl 0938.52003)], the author studies the absolute continuity of the \(r\)th curvature measure \(C_r(K,\cdot)\) of a convex body \(K\subset \mathbb{R}^d\) with interior points (w.r.t. \(C_{d-1}(K,\cdot)\), the \((d-1)\)-dimensional Hausdorff measure on the boundary of \(K)\). Based on a representation of the singular part of \(C_r(K,\cdot)\) from Part I, the absolute continuity of \(C_r(K,\cdot)\) on a Borel set \(\beta\subset\mathbb{R}^d\) is expressed in terms of generalized curvatures on the unit normal bundle of \(K\). For \(r=0\), the absolute continuity is then shown to be equivalent to a supporting property from inside by \(d\)-dimensional balls. This result is generalized to \(r\in\{1, \dots,d-2\}\) with the help of the Crofton formula for curvature measures, the corresponding supporting property from inside by \((d-r)\)-dimensional balls is then required for (almost) all sections \(K\cap E\) of \(K\) with affine \((d-r)\)-flats \(E\). A formulation in terms of touching flats \(E\) is also given. The main tool for the application of the integral geometric results is a theorem, which relates the absolute continuity of curvature measures of \(K\) with the corresponding properties of sections \(K\cap E\). The interesting paper contains also various analogous results for surface area measures.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 1384887
zbMATH Keywords: surface area measure
zbMATH Keywords: absolute continuity
zbMATH Keywords: curvature measure
zbMATH Keywords: convex body
zbMATH Keywords: Crofton formula
reviewed by: Wolfgang Weil
MaRDI profile type: Publication
full work available at URL: pl00004765
OpenAlex ID: W2129296215
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description: scientific article; zbMATH DE number 1384887