照亮本原多面体
Illuminating Primitive Polytopes
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中文总结 AI 辅助
研究本原凸\(d -\)多面体的照明问题，通过证明得出任何此类多面体最多可用\(2^d\)个方向照亮，非\(d -\)立方体线性图像的所需方向更少。
中文摘要 AI 辅助
一个凸\(d\)维多面体可定义为\(\mathbb{E}^d\)中具有内部的闭半空间的有界交集。若去掉任何一个半空间会使交集无界，则该多面体是本原的。本文证明了本原多面体的照明猜想：任何本原凸\(d -\)多面体\(P\)最多可用\(2^d\)个方向照亮；若\(P\)不是\(d -\)立方体的线性图像，则所需方向更少。
英文摘要
A convex $d$-dimensional polytope may be defined as a bounded intersection of closed halfspaces in $\mathbb{E}^d$ with interior. A polytope is primitive if omitting any halfspace renders the intersection unbounded. In this paper, we prove the Illumination Conjecture for the primitive polytopes: any primitive convex $d$-polytope $P$ can be illuminated with at most $2^d$ directions; even fewer if $P$ is not a linear image of a $d$-cube.
