Spherical coverings and X-raying convex bodies of constant width
A. Bondarenko, A. Prymak, D. Radchenko
TL;DR
The paper addresses the X-ray and illumination conjectures for convex bodies of constant width by constructing explicit origin-symmetric spherical coverings with radius $r=\arccos\sqrt{\tfrac{n-1}{2n}}$ for $5\le n\le15$, achieving $w_n<2^n$ and hence $X_n^w\le 2^{n-1}$ in these dimensions. It introduces a practical polar-dual framework to compute covering radii and builds symmetric, permutation-invariant generating sets that yield concrete upper bounds $w_n$; the $n=8$ case leverages the $E_8$ lattice and is extended to all $n$ in the range. The results resolve the previously outstanding cases $7\le n\le15$ and provide a scalable computational approach (via SageMath) for evaluating coverings. This work strengthens connections between spherical coverings, X-raying, and illumination in convex geometry and offers actionable constructions for higher-dimensional constant-width bodies.
Abstract
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.