论文标题
球形覆盖物和X射线凸形体的恒定宽度
Spherical coverings and X-raying convex bodies of constant width
论文作者
论文摘要
K. Bezdek和Gy。 KISS表明，在$ \ mathbb {e}^n $中，单位球的起源对称覆盖物的存在最多最多是$ 2^n $一致的球形帽，半径不超过$ \ arccos \ arccos \ arccos \ sqrt {\ frac {\ frac {n-1}} $ \ mathbb {e}^n $，并以$ 4 \ le n \ le 6 $构建了此类覆盖物。在这里，我们以$ 5 \ le n \ le 15 $的价格提供了不到$ 2^n $帽子的这种结构。 对于$ \ mathbb {e}^n $，o。〜schramm中任何恒定宽度的任何凸体的照明编号，证明了一个上限的估计值，订单$（3/2）^{n/2} $的指数增长。特别是，该估计值小于$ 3 \ cdot 2^{n-2} $，对于$ n \ ge 16 $，确认了上述凸出构体的凸起的凸起的猜想。因此，我们的结果解决了未偿还的案件$ 7 \ le n \ le 15 $。 我们还展示了如何在计算机上有效地在球体上设置给定离散点的覆盖半径。
K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray conjecture and the illumination conjecture for convex bodies of constant width in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we give such constructions with fewer than $2^n$ caps for $5\le n\le 15$. For the illumination number of any convex body of constant width in $\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$ for $n\ge 16$, confirming the above mentioned conjectures for the class of convex bodies of constant width. Thus, our result settles the outstanding cases $7\le n\le 15$. We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer.
