Asymptotic shape of a random polytope in a convex body |
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Authors: | N. Dafnis A. Giannopoulos A. Tsolomitis |
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Affiliation: | aDepartment of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece;bDepartment of Mathematics, University of the Aegean, Karlovassi 832 00, Samos, Greece |
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Abstract: | Let K be an isotropic convex body in and let Zq(K) be the Lq-centroid body of K. For every N>n consider the random polytope KN:=conv{x1,…,xN} where x1,…,xN are independent random points, uniformly distributed in K. We prove that a random KN is “asymptotically equivalent” to Z[ln(N/n)](K) in the following sense: there exist absolute constants ρ1,ρ2>0 such that, for all and all N N(n,β), one has:- (i) KN
c(β)Zq(K) for every q ρ1ln(N/n), with probability greater than 1−c1exp(−c2N1−βnβ). - (ii) For every q
ρ2ln(N/n), the expected mean width of KN is bounded by c3w(Zq(K)). As an application we show that the volume radius |KN|1/n of a random KN satisfies the bounds for all N exp(n). Keywords: Convex body; Isotropic body; Isotropic constant; Random polytope; Centroid bodies; Mean width; Volume radius |
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Keywords: | Convex body Isotropic body Isotropic constant Random polytope Centroid bodies Mean width Volume radius |
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