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DCG

2008

2008

In [GL] it was shown that a polytope with few vertices is far from being symmetric in the Banach-Mazur distance. More precisely, it was shown that Banach-Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach-Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in [BL]. Furthermore, we give the affirmative answer to a conjecture by K. Bezdek [B3] on the quantitative illumination problem.

Added |
10 Dec 2010 |

Updated |
10 Dec 2010 |

Type |
Journal |

Year |
2008 |

Where |
DCG |

Authors |
E. D. Gluskin, A. E. Litvak |

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