Some Generalizations of the Shadow Problem in the Lobachevsky Space

Abstract

We consider the problem of shadow in the Lobachevsky space. This problem can be treated as the problem of finding conditions guaranteeing that points belong to the generalized convex hull of a family of sets. We determine the limit values of the parameters for which the same configurations of balls guarantee that a point belongs to the generalized convex hull of balls in the Euclidean and hyperbolic spaces. Parallel with families of balls, we consider families of horoballs, as well as certain combinations of balls and horoballs.