Localization results for Minkowski contents

摘要

It was shown recently that the Minkowski content of a bounded set $A$ in$mathbb{R}^d$ with volume zero can be characterized in terms of the asymptoticbehaviour of the boundary surface area of its parallel sets $A_r$ as theparallel radius $r$ tends to $0$. Here we discuss localizations of suchresults. The asymptotic behaviour of the local parallel volume of $A$ relativeto a suitable second set $Omega$ can be understood in terms of the suitablydefined local surface area relative to $Omega$. Also a measure version of thisrelation is shown: Viewing the Minkowski content as a locally determinedmeasure, this measure can be obtained as a weak limit of suitably rescaledsurface measures of close parallel sets. Such measure relations had beenobserved before for self-similar sets and some self-conformal sets in$mathbb{R}^d$. They are now established for arbitrary closed sets, includingeven the case of unbounded sets. The results are based on a localization of Stach'o's famous formula relatingthe boundary surface area of $A_r$ to the derivative of the volume function at$r$.