Some remarks about system of balls that generate shadow at point

TL;DR

This paper investigates the minimal number of disjoint balls in multi-dimensional space needed to cast a shadow at a fixed point, introducing new properties and constructions for such systems.

Contribution

It presents new properties of disjoint balls in space and constructs systems of n+1 equal-radius balls in n-dimensional space that generate shadows at a point.

Findings

Established properties of disjoint balls generating shadows

Constructed systems of n+1 equal-radius balls in n-dimensional space

Analyzed minimal configurations for shadow generation

Abstract

Problems, related to the determination of the minimal number of balls that generate a shadow at a fixed point in the multi-dimensional Euclidean space $ \mathbb{R}^n $, are considered in present work. Here, the statement "a system of balls generate shadow at a point" means that any line passing through the point intersects at least one ball of the system. New properties of pairwise-disjoint balls that do not contain a fixed point inside of a sphere in space $\mathbb{R}^3$, centered on the sphere, and generate shadow at the point are established. And a system of $n+1$ pairwise-disjoint balls with equal radii in $\mathbb{R}^n$, $ n\ge 3$, that do not contain a fixed point of the space and generate shadow at the point is constructed in the work.