Localization Results For Minkowski Contents

It was shown recently that the Minkowski content of a bounded set A in Rd with volume zero can be characterized in terms of the asymptotic behaviour of the boundary surface area of its parallel sets Ar 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 local Minkowski content can be obtained as a weak limit of suitably rescaled surface measures of close parallel sets (local S-content). Such measure relations had been observed before for self-similar and some self-conformal sets. They are now established for arbitrary closed sets, including even the case of unbounded sets. The results are based on a localization of Stacho's famous formula relating the boundary surface area of Ar to the derivative of the volume function at r.