The problem of shadow for domains in Euclidean spaces

Abstract

The problem of shadow generalized onto domains of the space ℝⁿ, n ≤ 3, is investigated. The problem consists in the determination of the minimal number of balls satisfying some conditions such that every line passing through the given point intersects at least one ball of the collection. We have proved that it is sufficient to have four (two) mutually nonoverlapping closed or open balls in order to generate a shadow at every given point of any domain of the space ℝ³ (ℝ²). They do not include the point, and their centers lie on the domain boundary.