Every $3$-dimensional convex body can be covered by $14$ smaller homothetic copies

TL;DR

This paper proves that any 3D convex body can be covered by 14 smaller homothetic copies, improving previous bounds and using a novel discretization and computational approach.

Contribution

It introduces a new discretization technique and computational method to reduce the covering problem to linear programming feasibility checks.

Findings

Every 3D convex body can be covered by 14 smaller homothetic copies.

The approach reduces the problem to linear programs with rational coefficients.

The method improves upon the previous bound of 16 copies.

Abstract

We show that every $3$-dimensional convex body can be covered by $14$ smaller homothetic copies. The previous result was $16$ copies established by Papadoperakis in 1999, while a conjecture by Hadwiger is $8$. We modify Papadoperakis's approach and develop a discretization technique that reduces the problem to verification of feasibility of a number of linear programs with rational coefficients, which is done with computer assistance using exact arithmetic.