Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45-48, [2002]) 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 (Bezdek, Litvak in Adv. Math. 215:626-641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59-69, [2006]) on the quantitative illumination problem.