This is an announcement for the paper “Geometry of random sections of isotropic convex bodies” by Apostolos Giannopouloshttp://arxiv.org/find/math/1/au:+Giannopoulos_A/0/1/0/all/0/1, Labrini Hionihttp://arxiv.org/find/math/1/au:+Hioni_L/0/1/0/all/0/1, Antonis Tsolomitishttp://arxiv.org/find/math/1/au:+Tsolomitis_A/0/1/0/all/0/1.
Abstract: Let K be an isotropic symmetric convex body in ℝn. We show that a subspace F∈Gn,n−k of codimension k=γn, where γ∈(1/n√,1), satisfies K∩F⊆(cγ−1n√LK)Bn2∩F with probability greater than 1−exp(−n√). This implies that a random U∈O(n) satisfies K∩U(K)⊆(c1n√LK)Bn2 with probability greater than 1−e−n, where c1>0 is an absolute constant. Using a different method we study the same question for the Lq-centroid bodies Zq(μ) of an isotropic log-concave probability measure μ on ℝn. For every 1⩽q⩽n and γ∈(0,1) we show that a random subspace F∈Gn,γn satisfies Zq(μ)∩F⊆c2(γ)q√Bn2∩F; this implies that a random U∈O(n) satisfies Zq(μ)∩U(Zq(μ))⊆c3q√Bn2. Finally, we show that if an isotropic symmetric convex body has maximal isotropic constant then a random n/2-dimensional section K∩F of K has isotropic constant LK∩F⩽c4, where c4>0 is an absolute constant, and that K∩U(K)⊆c5n√Bn2 for a random U∈O(n).
The paper may be downloaded from the archive by web browser from URL
http://arxiv.org/abs/1601.02254
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