关于一个 Gallai 型问题以及刺球与帽体的照明
On a Gallai-type problem and illumination of spiky balls and cap bodies
AI总结：
本文研究两两边交球族的 Gallai 型刺穿问题，并将结果用于刺球与帽体照明，显著改进高维上下界。
摘要：
我们证明，在 $\mathbb{E}^n$ 中，任意有限的两两边交的球族都可以被 $(\sqrt{3/2}+o(1))^n$ 个点击穿，这改进了此前已知的 $(2+o(1))^n$ 估计。作为推论，这意味着 $\mathbb{E}^n$ 中任意可 2-照明的刺球都可以由 $(\sqrt{3/2}+o(1))^n$ 个方向照明。对于凸刺球（即帽体）的照明数，我们依据某些相关球面码和覆盖的大小给出了一个上界。在高维情形下，这得到 $1.19851^n$ 的上界；相比之下，此前的 $(\sqrt{2}+o(1))^n$ 上界仅对中心对称帽体成立。我们还证明了上述三个问题的下界均为 $(\tfrac{2}{\sqrt{3}}-o(1))^n$。
英文摘要：
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.
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