On volume and surface area of parallel sets Jan RatajSteffen Winter 28A7528A8052A2060D05parallel setsurface areaMinkowski contentMinkowski dimensionself-similar setrandom set The $r$-parallel set to a set $A$ in a Euclidean space consists of all points with distance at most $r$ from $A$. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities as $r$ tends to $0$. We show, for instance, that in general, the existence of a (suitably rescaled) limit of the surface area implies the existence of the corresponding limit for the volume, known as the Minkowski content. A full characterisation is obtained for the case of self-similar fractal sets. Applications to stationary random sets are discussed as well, in particular, to the trajectory of the Brownian motion. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4165 10.1512/iumj.2010.59.4165 en Indiana Univ. Math. J. 59 (2010) 1661 - 1686 state-of-the-art mathematics http://iumj.org/access/