Characterization of Minkowski measurability in terms of surface area

The rr-parallel set to a set AA in Euclidean space consists of all points with distance at most rr from AA. Recently, the asymptotic behaviour of volume and surface area of the parallel sets as rr tends to 00 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 SS-content. In particular, we show that a set is Minkowski measurable if and only if it is SS-measurable, i.e. if and only if its SS-content is positive and finite, and that positivity and finiteness of the lower and upper Minkowski contents imply the same for the SS-contents and vice versa. The results are formulated in the more general setting of Kneser functions. Furthermore, the relations between Minkowski and SS-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.