Hadwiger's conjecture for cap bodies

Abstract

Hadwiger's covering conjecture states that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic translates, with $2^n$ copies required only for affine images of the $n$-cube. In this note, we confirm Hadwiger's conjecture for the class of cap bodies in all dimensions, bridging recently established cases of $n=3$ and large $n$. The proof uses probabilistic techniques and, for $4\le n\le 15$, computer-assisted linear programming.

14 pages