Volume on stratified sets
type: Item
instance of: scholarly article
title: Volume on stratified sets
zbMATH Open document ID: 0612.58004
DOI: 10.1007/BF01760818
author: Massimo Ferrarotti
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1986-01-01T00:00:00Z
review text: Let \(W=\cup S\) (S\(\in {\mathcal S})\) be a stratification of a closed subset W of a Riemannian manifold M and \(E\subset W\) be a \(\sigma\)-compact subset. Let \({\mathcal N}\) be the direct system of open neighbourhoods of the (p-1)- dimensional skeleton of the stratification, where \(p=\dim W=\max \dim S\) (S\(\in {\mathcal S})\). The volume of E is defined by \(Vol(E)=dir \lim Vol(E- N)\) (N\(\in {\mathcal N})\) and does not depend on the auxiliary stratification. We say that W is v-regular if, for each relatively compact subset K in M, the estimate Vol \((W\cap B_ r)\leq c(K)r^ p\) is true (with appropriate c(K)) for each ball \(B_ r\), \(\bar B{}_ r\subset K\). Theorems: If a subset \(W\subset M\) admits a Whitney-regular stratification with strata in dimensions p and p-1 only, then W is v- regular. Subanalytic subsets W of M are always v-regular but in general, the above skeleton condition cannot be omitted.
Mathematics Subject Classification ID: 58A35
Mathematics Subject Classification ID: 58C35
Mathematics Subject Classification ID: 28C99
zbMATH DE Number: 3990424
zbMATH Keywords: stratification
zbMATH Keywords: volume
zbMATH Keywords: Whitney regularity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3990424