Hadwiger's conjecture for cap bodies

TL;DR

This paper proves Hadwiger's covering conjecture for cap bodies across all dimensions using probabilistic methods and computer-assisted integer linear programming for moderate dimensions, advancing the understanding of convex body coverings.

Contribution

It confirms Hadwiger's conjecture for cap bodies in all dimensions, bridging previous partial results with new probabilistic and computational techniques.

Findings

Conjecture holds for cap bodies in all dimensions.

Probabilistic techniques are effective in the proof.

Computer-assisted ILP aids in moderate dimensions.

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.