Illumination of convex bodies with many symmetries

TL;DR

The paper proves that convex bodies with high symmetry in high dimensions, other than cubes, can be illuminated with fewer than 2^n light sources, confirming a longstanding conjecture for certain symmetric norm balls.

Contribution

It establishes the illumination conjecture for unit balls of 1-symmetric norms in high dimensions, showing they require fewer than 2^n lights if not cubes.

Findings

Convex bodies with symmetric properties can be illuminated with fewer than 2^n sources.

The result confirms the conjecture for large-dimensional symmetric norm balls.

Cubes are the only bodies requiring the maximum number of lights.

Abstract

Let $n\geq C$ for a large universal constant $C>0$, and let $B$ be a convex body in $R^n$ such that for any $(x_1,x_2,\dots,x_n)\in B$, any choice of signs $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n\in\{-1,1\}$ and for any permutation $\sigma$ on $n$ elements we have $(\varepsilon_1x_{\sigma(1)},\varepsilon_2x_{\sigma(2)},\dots,\varepsilon_nx_{\sigma(n)})\in B$. We show that if $B$ is not a cube then $B$ can be illuminated by strictly less than $2^n$ sources of light. This confirms the Hadwiger--Gohberg--Markus illumination conjecture for unit balls of $1$-symmetric norms in $R^n$ for all sufficiently large $n$.