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arXiv · 1606.08976

Illumination of convex bodies with many symmetries

Konstantin Tikhomirov
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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 $σ$ on $n$ elements we have $(\varepsilon_1x_{σ(1)},\varepsilon_2x_{σ(2)},\dots,\varepsilon_nx_{σ(n)})\in B$. 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 $n$.

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Konstantin Tikhomirov. 2016-10-05. Illumination of convex bodies with many symmetries. https://doi.org/10.1112/s0025579316000292

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