Localization results for Minkowski contents

Abstract

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 asymptotic behaviour of the boundary surface area of its parallel sets $A_r$ as the parallel radius $r$ tends to $0$. Here we discuss localizations of such results. The asymptotic behaviour of the local parallel volume of $A$ relative to a suitable second set $\Omega$ can be understood in terms of the suitably defined local surface area relative to $\Omega$. Also a measure version of this relation is shown: Viewing the Minkowski content as a locally determined measure, this measure can be obtained as a weak limit of suitably rescaled surface measures of close parallel sets. Such measure relations had been observed before for self-similar sets and some self-conformal sets in $\mathbb{R}^d$. They are now established for arbitrary closed sets, including even the case of unbounded sets. The results are based on a localization of Stach\'o's famous formula relating the boundary surface area of $A_r$ to the derivative of the volume function at $r$.