Spherical coverings and X-raying convex bodies of constant width

Abstract

Bezdek and Kiss showed that existence of origin-symmetric coverings of unit sphere in En by at most 2n congruent spherical caps with radius not exceeding arccosn−12n−−−√ implies the X-ray conjecture and the illumination conjecture for convex bodies of constant width in En , and constructed such coverings for 4≤n≤6 . Here, we give such constructions with fewer than 2n caps for 5≤n≤15 .

For the illumination number of any convex body of constant width in En , Schramm proved an upper estimate with exponential growth of order (3/2)n/2 . In particular, that estimate is less than 3⋅2n−2 for n≥16 , confirming the abovementioned conjectures for the class of convex bodies of constant width. Thus, our result settles the outstanding cases 7≤n≤15 .

We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer.