Characterization of Minkowski measurability in terms of surface area

Abstract

The r-parallel set to a set A in Euclidean space consists of all points with distance at most r from A. Recently, the asymptotic behaviour of volume and surface area of the parallel sets as r tends to 0 has been studied and some general results regarding their relations have been established. Here we complete the picture regarding the resulting notions of Minkowski content and S-content. In particular, we show that a set is Minkowski measurable if and only if it is S-measurable, i.e. if and only if its S-content is positive and finite, and that positivity and finiteness of the lower and upper Minkowski contents imply the same for the S-contents and vice versa. The results are formulated in the more general setting of Kneser functions. Furthermore, the relations between Minkowski and S-contents are studied for more general gauge functions. The results are applied to simplify the proof of the Modified Weyl–Berry conjecture in dimension one.