Spherical coverings and X-raying convex bodies of constant width

• Published in: Canadian mathematical bulletin 2022-12, Vol.65 (4), p.860-866
• Main Authors: Bondarenko, Andriy, Prymak, Andriy, Radchenko, Danylo
• Format: Article
• Language: English
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439521001016

Bezdek and 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 ${\mathbb {E}}^n$ , 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 abovementioned 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.