Minkowski content; Kneser function; Brownian motion; Wiener sausage
We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset \mathbb R^d$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in $\mathbb R^d$, $d\geq 3$.
 Berezhkovskii, A. M., Makhnovskii, Yu. A., Suris, R. A.: Wiener sausage volume moments
. J. Stat. Phys. 57 (1989), 333-346. DOI 10.1007/BF01023647
| MR 1031416
 Stachó, L. L.: On the volume function of parallel sets
. Acta Sci. Math. 38 (1976), 365-374. MR 0442202
| Zbl 0342.52014