论文标题
照亮尖刺的球和帽子
Illuminating spiky balls and cap bodies
论文作者
论文摘要
具有外部点的球的凸壳称为尖峰（或盖）。有限的球尖峰的结合称为尖峰球。如果一个尖刺的球是凸的，那么我们将其称为帽子。在本说明中，我们在上面绑定了$ 2 $倒数的尖刺球以及中央对称的帽子的上限。特别是，我们证明了在足够大的尺寸中的中央对称帽子的照明猜想，这表明在Euclidean $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d \ degeq 20 $的$ <2^d $方向都可以照亮。此外，我们以1美元的对称帽子的身份加强后者的结果。
The convex hull of a ball with an exterior point is called a spike (or cap). A union of finitely many spikes of a ball is called a spiky ball. If a spiky ball is convex, then we call it a cap body. In this note we upper bound the illumination numbers of $2$-illuminable spiky balls as well as centrally symmetric cap bodies. In particular, we prove the Illumination Conjecture for centrally symmetric cap bodies in sufficiently large dimensions by showing that any $d$-dimensional centrally symmetric cap body can be illuminated by $<2^d$ directions in Euclidean $d$-space for all $d\geq 20$. Furthermore, we strengthen the latter result for $1$-unconditionally symmetric cap bodies.
