Illumination of convex bodies with many symmetries

Abstract

Let $n\ge C$ 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 ${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n\in \{-1,1\}$ and for any permutation $\sigma$ on $n$ elements we have $({\epsilon }_1{x}_{\sigma (1)},{\epsilon }_2{x}_{\sigma (2)},\dots ,{\epsilon }_n{x}_{\sigma (n)})\in B$. 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 $n$.

References in corpus (1)

• A Spiky Ball

Cited by in corpus (2)

• On the illumination of centrally symmetric cap bodies in small dimensions
• Covering by homothets and illuminating convex bodies