Inserted: 26 sep 2007
We find conditions ensuring the existence of the one-sided Minkowski content for $d$-dimensional closed sets in $*R*^d$, in connection with regularity properties of their boundaries. Moreover, we provide a class of sets stable under finite unions for which the one-sided Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class. We find analogous conditions, stable under finite unions as well, for the existence of the mean one-sided Minkowski content of random closed sets. Finally, an application to birth-and-growth stochastic processes is briefly discussed.