A Remark on Vertex Index of the Convex Bodies

Abstract

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.