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On the ψ₂-behaviour of linear functionals on isotropic convex bodies

G. Paouris — 2005

Studia Mathematica

The slicing problem can be reduced to the study of isotropic convex bodies K with d i a m ( K ) c n L K , where L K is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that | | · , θ | | ψ C L K for all θ in a subset U of S n - 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 m a x θ S n - 1 | | · , θ | | ψ c n L K . In a different direction, we show that good average ψ₂-behaviour of linear functionals on an isotropic...

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