On the ψ 2 -behaviour of linear functionals on isotropic convex bodies
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The slicing problem can be reduced to the study of isotropic convex bodies K with diam(K) ≤ c√nLK, where LK is the isotropic constant. We study the ψ2-behaviour of linear functionals on this class of bodies. It is proved that ||〈·,θ〉||ψ2 ≤ CLK for all θ in a subset U of Sn-1 with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that maX θ∈Sn-1 ||〈·,θ〉||ψ2 ≥ c4√nLK. In a different direction, we show that good average ψ2-behaviour of linear functionals on an isotropic convex body implies very strong dimension-dependent concentration of volume inside a ball of radius r ≃ √nLK.
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