On the shadow problem for domains in the Euclidean spaces

TL;DR

This paper investigates the minimal number of non-overlapping balls needed to ensure every line through a point in a domain in 2D or 3D intersects at least one ball, establishing that four balls suffice.

Contribution

It proves that four non-overlapping balls with centers on the boundary are sufficient to generate a shadow at any point in a domain in D and 2D, generalizing the shadow problem.

Findings

Four balls suffice to generate the shadow at any point in D and 2D domains.

The balls are non-overlapping, can be open or closed, and do not contain the point.

Centers of the balls are on the boundary of the domain.

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.