On the vertex index of convex bodies

Karoly Bezdek; Alexander E. Litvak

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arXiv · 1110.4334

On the vertex index of convex bodies

Karoly Bezdek·Alexander E. Litvak

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Abstract

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2^d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. Also, we provide sharp estimates in dimensions 2 and 3.

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Karoly Bezdek, Alexander E. Litvak. 2011-10-19. On the vertex index of convex bodies. https://arxiv.org/abs/1110.4334

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