Isomorphic properties of intersection bodies
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We study isomorphic properties of two generalizations of intersection bodies  the class Ikn of kintersection bodies in Rn and the class BPkn of generalized kintersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n1 then the outer volume ratio distance from K to the class BPkn can be estimated by. o.v.r.(K,BPkn):=inf{(CK)1n:C∈BPkn,K≤C}≤cnklogenk, where c>0 is an absolute constant. Next we prove that if K is a symmetric convex body in Rn, 1≤k≤n1 and its kintersection body Ik(K) exists and is convex, then. dBM(Ik(K),B2n)≤c(k), where c(k) is a constant depending only on k, dBM is the BanachMazur distance, and B2n is the unit Euclidean ball in Rn. This generalizes a wellknown result of Hensley and Borell. We conclude the paper with volumetric estimates for kintersection bodies. © 2011 Elsevier Inc.
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Koldobsky, A., Paouris, G., & Zymonopoulou, M.
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