Geometric Aspects of Functional Analysis pp 255-265 | Cite as

# A Remark on Vertex Index of the Convex Bodies

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## Abstract

The vertex index of a symmetric convex body \(\mathbf{K} \subset {\mathbb{R}}^{n}\), vein(which is asymptotically sharp. We also show that the estimate obtained in [Bezdek, Litvak, Adv. Math.

**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$$\mathrm{vein}(\mathbf{K}) \leq 24{n}^{3/2},$$

$$\frac{{n}^{3/2}} {\sqrt{2\pi e}\ \mbox{ ovr}(\mathbf{K})} \leq \mathrm{vein}(\mathbf{K}),$$

**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.## Keywords

Convex Hull Convex Body Absolute Constant Logarithmic Term Euclidean Ball
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgements

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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