A Remark on Vertex Index of the Convex Bodies

Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)


The vertex index of a symmetric convex body \(\mathbf{K} \subset {\mathbb{R}}^{n}\), vein(K), was introduced in [Bezdek, Litvak, Adv. Math. 215, 626–641 (2007)]. Bounds on the vertex index were given in the general case as well as for some basic examples. In this note we improve these bounds and discuss their sharpness. We show that
$$\mathrm{vein}(\mathbf{K}) \leq 24{n}^{3/2},$$
which is asymptotically sharp. We also show that the estimate
$$\frac{{n}^{3/2}} {\sqrt{2\pi e}\ \mbox{ ovr}(\mathbf{K})} \leq \mathrm{vein}(\mathbf{K}),$$
obtained in [Bezdek, Litvak, Adv. Math. 215, 626–641 (2007)] (here ovr(K) denotes the outer volume ratio of K), is not always sharp. Namely, we construct an example showing that there exists a symmetric convex body K which simultaneously has large outer volume ratio and large vertex index. Finally, we improve the constant in the latter bound for the case of the Euclidean ball from \(\sqrt{2\pi e}\) to \(\sqrt{3}\), providing a completely new approach to the problem.


Convex Hull Convex Body Absolute Constant Logarithmic Term Euclidean Ball 
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Part of this research was conducted while the second named author participated in the Thematic Program on Asymptotic Geometric Analysis at the Fields Institute in Toronto in Fall 2010. He thanks the Institute for the hospitality. His research partially supported by the E.W.R. Steacie Memorial Fellowship.


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Sackler Faculty of Exact Sciences, Department of MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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