On the vertex index of convex bodies |
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Authors: | K Bezdek AE Litvak |
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Institution: | a Department of Mathematics and Statistics, 2500 University drive N.W., University of Calgary, AB, Canada, T2N 1N4 b Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada, T6G 2G1 |
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Abstract: | We introduce the vertex index, vein(K), of a given centrally symmetric convex body K⊂Rd, 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 d2 smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K⊂Rd one has |
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Keywords: | primary 46B07 46B09 52A secondary 51M16 53A55 |
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