On the shadow problem for domains in the Euclidean spaces

Abstract

In the present work, the problem about shadow, generalized on domains of space $\mathbb{R}^n$, $n\le 3$, is investigated. Here the shadow problem means to find the minimal number of balls satisfying some conditions an such that every line passing through the given point intersects at least one ball of the collection. It is proved that to generate the shadow at every given point of any domain of the space $\mathbb{R}^3$ ($\mathbb{R}^2$) with collection of mutually non-overlapping closed or open balls which do not hold the point and with centers on the boundary of the domain, it is sufficient to have four balls.