论文标题
在小维度的中央对称帽子的照明上
On the illumination of centrally symmetric cap bodies in small dimensions
论文作者
论文摘要
euclidean space in euclidean space $ k $的照明编号$ i（k）$是完全阐明凸形边界的最小方向。一个球的帽子主体$ k_c $是欧几里得球的凸壳，在连接这些点的每个段相交的条件下，球外部的一组可数点。本文的主要结果是$ i（k_c）\ leq6 $的中央对称帽子$ \ mathbb {e}^3 $，以及$ i（k_c）\ leq 8 $的无条件对称性帽子的$ \ mathbbbb中的$ \ mathbb {e e}^4 $。
The illumination number $I(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$ is the smallest number of directions that completely illuminate the boundary of a convex body. A cap body $K_c$ of a ball is the convex hull of a Euclidean ball and a countable set of points outside the ball under the condition that each segment connecting two of these points intersects the ball. The main results of this paper are the sharp estimates $I(K_c)\leq6$ for centrally symmetric cap bodies of a ball in $\mathbb{E}^3$, and $I(K_c)\leq 8$ for unconditionally symmetric cap bodies of a ball in $\mathbb{E}^4$.
