Summary: Let \(X_{1},X_{2},\dots\) be a sequence of independent identically distributed random closed subsets of a certain locally compact, Hausdorff, and separable space \(E\). For each random closed set \(Y\) we consider its avoidance functional \(Q_{Y}(F)\) equal to the probability that \(Y\) is disjoint with the closed subset \(F \subseteq E\). The purpose of this paper is to establish limit theorems for the random variables \(Q_{Y}(X_{1} \cup \cdots \cup X_{n})\). The results obtained are then applied for asymptotic analysis of the mean width of convex hulls generated by uniform samples on a multidimensional ball.