The minimum variance of a random set on a Euclidean space

We consider the set-valued variance of random sets (Kruse variance) in those cases where the outcomes of the random set are closed subsets of a (finite dimensional) Euclidean space. We prove necessary and sufficient conditions for the existence and uniqueness of a random selection with minimum variance (minimal selection). In addition, we characterize the critical point of the minimization problem. More specifically, we provide equations that characterize the expectation of the minimal selection, as well as some additional conditions that constrain such selection. Special attention is paid to two special families of closed convex random sets, namely random closed balls and random closed intervals, for which stronger conclusions are obtained.