Sciweavers

Share
DCG
2008

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

12 years 1 months ago
Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies
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.
E. D. Gluskin, A. E. Litvak
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2008
Where DCG
Authors E. D. Gluskin, A. E. Litvak
Comments (0)
books