Generalized convex envelopes of sets and the problem of shadow (Q905437)

The problem of shadow was posed by \textit{G. Khudaiberganov} in [On the homogeneous polynomially convex envelope of a union of balls (Russian). Moscow: VINTI. Manuscript No. 1772 (1982)]. It says: Which minimum number of pairwise disjoint closed balls in the Euclidean space \(\mathbb R^n\) with centers on the sphere \(S^{n-1}\) and of radii not exceeding that of the sphere is sufficient for any straight line passing through the sphere center to cross at least one of these balls. The main aim of the present paper is to survey the results related to the solution of the shadow problem and discuss a number of related problems.