Let $B$ ("black") and $W$ ("white") be disjoint compact test sets in $\mathbb{R}^d$ and consider the volume of all its simultaneous shifts keeping $B$ inside and $W$ outside a compact set $A\subset\mathbb{R}^d$. If the union $B\cup W$ is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of $A$ and the support functions of $B$ and $W$, provided that $A$ is regular enough (e.g. polyconvex). An analogous formula is obtained for the case when the conditions $B\subset A$ and $W\subset A^C$ are replaced with prescribed threshold volumes of $B$ in $A$ and $W$ in $A^C$.

Applications in stochastic geometry are discussed. Firstly, the hit distribution function of a random set with an arbitrary compact structuring element $B$ is considered. Its derivative at $0$ is expressed in terms of the rose of directions and $B$. An analogue result holds for the hit-or-miss function. Secondly, in a desing based setting, different random digitizations of a deterministic set $A$ are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of $A$ as the lattice distance converges to zero.