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ILLUMINATION OF CONVEX BODIES WITH MANY SYMMETRIES

Part of: General convexity Discrete geometry

Published online by Cambridge University Press:  16 February 2017

Konstantin Tikhomirov
Konstantin Tikhomirov*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton AB T6G 2G1, Canada email [email protected], [email protected] Department of Mathematics, Princeton University, Princeton NJ 08544, U.S.A.
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Abstract

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 $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{n}\in \{-1,1\}$ and for any permutation $\unicode[STIX]{x1D70E}$ on $n$ elements, we have $(\unicode[STIX]{x1D700}_{1}x_{\unicode[STIX]{x1D70E}(1)},\unicode[STIX]{x1D700}_{2}x_{\unicode[STIX]{x1D70E}(2)},\ldots ,\unicode[STIX]{x1D700}_{n}x_{\unicode[STIX]{x1D70E}(n)})\in B$. 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 classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 372 - 382
Copyright
Copyright © University College London 2017 

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