Spherical coverings and X-raying convex bodies of constant width

A. Bondarenko, A. Prymak, D. Radchenko

Published 2020-11-12Version 1

The main goal of this work is to give constructions of certain spherical coverings in small dimensions. 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 $4\le n\le 6$. We give such constructions with fewer than $2^n$ caps for $5\le n\le 15$. In combination with the known result for higher dimensions due to O. Schramm, this completely settles the above mentioned conjectures for convex bodies of constant width in all dimensions. We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer.