Problem of shadow and surface of constant curvature

In this paper we consider the problem of shadow in the Lobachevsky space. This problem can be considered as the establishment of conditions to ensure the membership of the points to the generalized convex hull of a family of sets. The boundary values of the parameters are determined for which the same configurations of balls ensure that the point belongs to the generalized convex hull of balls in Euclidean and hyperbolic spaces. In addition to balls, the article discusses families of horoballs, as well as combinations of balls and horoballs. The article shows how the Euclidean surfaces of revolution of constant negative curvature are connected with tangent cones to the horospheres of the Lobachevsky space.