Hadwiger's conjecture for cap bodies

Abstract

Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an external point is called a spike. The union of finitely many spikes of a ball is a cap body if it is a convex set. In this note, we confirm the 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 additionally, for moderate dimensions $4\le n \le 15$, integer linear programming performed with computer assistance.