Volume on stratified sets
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.
scientific article; zbMATH DE number 3990424
scientific article