On a Gallai-type problem and illumination of spiky balls and cap bodies

TL;DR

This paper improves bounds on the number of points or directions needed to pierce or illuminate high-dimensional intersecting balls and cap bodies, advancing understanding of geometric covering and illumination problems.

Contribution

It provides new upper bounds for piercing and illumination numbers of intersecting balls and cap bodies in high dimensions, refining previous exponential estimates.

Findings

Piercing number improved to approximately (√3/2)^n

Upper bound for illumination of spiky balls is about (√3/2)^n

Lower bounds established at about (2/√3)^n

Abstract

We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above.