TY - CHAP

T1 - A remark on vertex index of the convex bodies

AU - Gluskin, Efim D.

AU - Litvak, Alexander E.

N1 - Funding Information:
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.

PY - 2012

Y1 - 2012

N2 - The vertex index of a symmetric convex body K Rn, 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 which is asymptotically sharp. We also show that the estimate 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 to , providing a completely new approach to the problem.

AB - The vertex index of a symmetric convex body K Rn, 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 which is asymptotically sharp. We also show that the estimate 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 to , providing a completely new approach to the problem.

UR - http://www.scopus.com/inward/record.url?scp=84865323019&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-29849-3_14

DO - 10.1007/978-3-642-29849-3_14

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SN - 978-3-642-29848-6

T3 - Lecture Notes in Math.

SP - 255

EP - 265

BT - Geometric aspects of functional analysis

A2 - Klartag, Bo'az

A2 - Mendelson, Shahar

A2 - Milman, Vitali D.

PB - Springer Heidelberg

ER -