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.

We study the farthest-point distance function, which measures the distance from $z\,\in \,\mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane.

The logarithm of this distance is subharmonic as a function of $z$, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure ${{\sigma }_{E}}$ has many interesting properties that reflect the topology and geometry of the compact set $E$. We prove ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$-gon for some odd $n$. Also we show ${{\sigma }_{E}}(E)\,=\,\frac{1}{2}$ for smooth convex sets of constant width. We conjecture ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for all $E$.