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

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.