A remark on mixed curvature measures for sets with positive reach
type: Item
instance of: scholarly article
title: A remark on mixed curvature measures for sets with positive reach
zbMATH Open document ID: 1008.53060
author: Jan Rataj
author: Martina Zähle
published in: Beiträge zur Algebra und Geometrie
publication date: 2002-08-20T00:00:00Z
full work available at URL: https://eudml.org/doc/122480
full work available at URL: 13.html
review text: For any two subsets \(X,Y\subseteq \mathbb{R}^d\) with positive reach and for \(r,s\in \{0,1, \dots, d-1\}\), \(r+s\geq d\), the mixed curvature measures \(C_{r,s} (X,Y;\cdot)\) have been defined by the authors in [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], where, under a non-osculating condition, a translative integral formula involving \(C_{r,s} (X,Y;\cdot)\) was proved. In the paper under review the authors show, through an interesting example, that the non-osculating condition does not ensure the locally bounded variation of the mixed curvature measures. In order to have \(C_{r,s} (X,Y;\cdot)\) well defined they assume that certain nonnegative measures \(|\overline C |_{r,s} (X,Y;\cdot)\), \(0\leq r\), \(s\leq d-1\), \(r+s\geq d\) are locally finite, and look for sufficient conditions for the property. This is always satisfied if \(d\leq 3\), as well as for any pair of convex sets. Moreover, for any \(X,Y \subseteq \mathbb{R}^d\) with positive reach, \(|\overline C|_{r,s} (X,\rho Y; \cdot)\) is locally finite for \(\vartheta_d\)-almost all rotations \(\rho\in SO(d)\), where \(\vartheta_d\) is the rotation invariant probability measure on \(SO(d)\). Further, for any \(X\subseteq\mathbb{R}^d\) with positive reach and locally bounded tangential projections, and for any \(L\) in the Grassmannian \(G(d,j)\) of \(j\)-dimensional linear subspaces in \(\mathbb{R}^d\) \((0<j\leq d)\), the pair \(X,L\) satisfies the non-osculating condition and \(|\overline C|_{r,s} (X,L;\cdot)\) is locally finite.
reviewed by: Paolo Dulio
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 1784600
zbMATH Keywords: existence of mixed curvature measures
zbMATH Keywords: sets with positive reach
zbMATH Keywords: locally bounded variation
zbMATH Keywords: local boundedness
zbMATH Keywords: absolute curvature measures
MaRDI profile type: Publication
Recommended article: Mixed curvature measures of translative integral geometry
Recommended article: Q4532352
Recommended article: Measure-valued valuations and mixed curvature measures of convex bodies
Recommended article: Iterations of translative integral formulae and non-isotropic Poisson processes of particles
Recommended article: Translative and kinematic integral formulae concerning the convex hull operation
Recommended article: Absolute curvature measures. II
Recommended article: Convex curves, Radon transforms and convolution operators defined by singular measures
Recommended article: Generalized curvature measures and singularities of sets with positive reach
Recommended article: Q3761210
Recommended article: Mixed measures and functionals of translative integral geometry
description: scientific article; zbMATH DE number 1784600