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 the surface area of parallel sets as $r$ tends to 0 has been studied and some general results regarding their relations have been established. In this paper we complete this picture. In particular, we show that a set is Minkowski measurable if and only if it is S-measurable, i.e. if its S-content is positive and finite, and that positivity and finiteness of lower and upper Minkowski content implies 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 also applied to simplify the proof of the Modified Weyl-Berry conjecture in dimension one.