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Illumination of convex bodies with many symmetries. (English) Zbl 1376.52004

Summary: Let \(n\geqslant C\) for a large universal constant \(C>0\) and let \(B\) be a convex body in \(\mathbb{R}^{n}\) such that for any \((x_{1},x_{2},\ldots ,x_{n})\in B\), any choice of signs \(\varepsilon_{1},\varepsilon_{2},\ldots ,\varepsilon_{n}\in \{-1,1\}\) and for any permutation \(\sigma\) on \(n\) elements, we have \((\varepsilon_{1}x_{\sigma(1)},\varepsilon_{2}x_{\sigma(2)},\ldots ,\varepsilon_{n}x_{\sigma(n)})\in B\). 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 \(\mathbb{R}^{n}\) for all sufficiently large \(n\).

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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