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

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Paouris G.

Studia Mathematica

2005

tom 168

nr 3

285-299

The slicing problem can be reduced to the study of isotropic convex bodies K with $diam (K) ≤ c√n L_{K}$, where $L_{K}$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that $||⟨·,θ⟩||_{ψ₂} ≤ CL_{K}$ 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 $max_{θ∈S^{n-1}} ||⟨·,θ⟩||_{ψ₂} ≥ c∜n L_{K}$. 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 $r ≃ √n L_{K}$.

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1 Studia Mathematica

1 Paouris G.

1 2005

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