Spherical coverings and X-raying convex bodies of constant width

Bondarenko, Andriy ; Prymak, Andriy  ; Radchenko, Danylo

Canadian mathematical bulletin, Tome 65 (2022) no. 4, pp. 860-866

Résumé

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.

Détail

DOI : 10.4153/S0008439521001016

Mots-clés : Spherical covering radius, X-ray problem, illumination problem, convex bodies of constant width