Random cell complexes and generalised sets
type: Item
instance of: scholarly article
title: Random cell complexes and generalised sets
zbMATH Open document ID: 0656.60024
published in: The Annals of Probability
publication date: 1988-01-01T00:00:00Z
review text: The concepts of random (curved) cell complexes, generalised sets, and their mean sets in \(R^ d\) are introduced. Under stationarity conditions various relations between associated geometrical quantities are derived. A central equation is the following stochastic mean value version of an extension of the Euler polyhedron theorem \((k=0)\) to Lipschitz-Killing curvature measures: \[ c^ i_ k =\sum^{i}_{j=k}(-1)^{j-k} N^ jC^ j_ k,\quad 0\leq k\leq i\leq d-1. \] Here \(c^ i_ k\) denotes the k-th Lipschitz-Killing curvature density of the random i-skeleton, \(N^ j\) the mean number of j-cells per unit volume, and \(C^ j_ k\) the mean total k-th Lipschitz-Killing curvature of the typical j-cell. (The surface area densities \(c^ i_ i\) and the mean surface areas \(C^ j_ j\) are contained as margins.) In the case of random tessellations the formulas may be further specified and the known results for convex mosaics in \(R^ 2\), \(R^ 3\) are extended.
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 58A25
zbMATH DE Number: 4072017
zbMATH Keywords: random cell complexes
zbMATH Keywords: curvature measures
zbMATH Keywords: random tessellations
zbMATH Keywords: convex mosaics
author: Martina Zähle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991595
OpenAlex ID: W2018282010
DOI: 10.1214/AOP/1176991595
description: scientific article; zbMATH DE number 4072017