Expressing intrinsic volumes as rotational integrals
type: Item
instance of: scholarly article
title: Expressing intrinsic volumes as rotational integrals
zbMATH Open document ID: 1202.60018
author: Jérémy Auneau
author: Eva B. Vedel Jensen
published in: Advances in Applied Mathematics
publication date: 2010-05-21T00:00:00Z
review text: A rotational formula of Crofton-type is derived for intrinsic volumes of a compact subset \(X\subset\mathbb{R}^d\) of positive reach. Motivated by applications in local stereology, the authors address the problem of finding functional \(\alpha_{jk}\) defined on the section of \(X\) with a \(j\)-dimensional linear subspace, and whose rotational average is equal to the intrinsic volume \(V_{d-j+k}(X)\) of \(X\), for \(j=0,1,\dots ,d\), and \(k=0,1,\dots ,j\). Solutions are derived for general \(j,k\). A simplified version is obtained in the case \(k=j\), while, for \(k<j\), there is a more explicit formula if \(X\) is the compact closure of an open subset of \(\mathbb{R}^d\), and \(\partial X\) is a special \((d-1)\)-dimensional manifold of class \(C^2\). When \(k=j-1\), the result can be further simplified.
reviewed by: Paolo Dulio
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 5710830
zbMATH Keywords: geometric measure theory
zbMATH Keywords: integral geometry
zbMATH Keywords: rotational integral
zbMATH Keywords: Grassmann manifold
zbMATH Keywords: intrinsic volume
zbMATH Keywords: set of positive reach
zbMATH Keywords: stereology
MaRDI profile type: Publication
full work available at URL: j.aam.2009.11.010
OpenAlex ID: W1998246035
cites work: Q3158365
cites work: A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications
cites work: Q4134602
cites work: Integral geometry of tensor valuations
cites work: Local Stereology
cites work: A rotational integral formula for intrinsic volumes
DOI: 10.1016/J.AAM.2009.11.010
description: scientific article; zbMATH DE number 5710830