Bounds for Restricted Selections of Random Sets

We study constrained selection sets of random closed sets defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we characterize the restricted selection set and establish sharp bounds on the attainable ranges of means, medians, and event probabilities. In particular, we give conditions under which every value in the Aumann expectation range is realized as the mean of a measurable selection, and we obtain explicit formulas for the extremal expectations under median and higher-moment restrictions via rearrangement and convex-duality arguments. We further show that the selection set of any random compact convex set in $\R^d$ can be approximated in $L^1$ by selection sets of disjoint unions of random cubes, each of which decomposes coordinate-wise into one-dimensional interval selection problems. This gives us an approximation-based reduction of constrained selection problems for random compact convex sets in $\R^d$.