Cube is a strict local maximizer for the illumination number

TL;DR

This paper proves that convex bodies close to a cube in the Banach-Mazur metric can be illuminated with fewer light sources than the general bound, specifically $2^n-1$, unless they are parallelotopes.

Contribution

It establishes a stability result for the illumination number near the cube, showing fewer sources suffice for bodies close to the cube but not parallelotopes.

Findings

Convex bodies near the cube require at most $2^n-1$ light sources for illumination.

Bodies close to the cube but not parallelotopes can be covered by $2^n-1$ smaller homothetic copies.

The result supports the conjecture for bodies close to the cube in the Banach-Mazur metric.

Abstract

It was conjectured by Levi, Hadwiger, Gohberg and Markus that the boundary of any convex body in ${\mathbb R}^n$ can be illuminated by at most $2^n$ light sources, and, moreover, $2^n-1$ light sources suffice unless the body is a parallelotope. We show that if a convex body is close to the cube in the Banach-Mazur metric, and it is not a parallelotope, then indeed $2^n-1$ light sources suffice to illuminate its boundary. Equivalently, any convex body sufficiently close to the cube, but not isometric to it, can be covered by $2^n-1$ smaller homothetic copies of itself.