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

G. Paouris

G. Paouris

Studia Mathematica, Tome 166 (2005), p. 285-299 / Harvested from The Polish Digital Mathematics Library

Résumé

The slicing problem can be reduced to the study of isotropic convex bodies K with $d i a m (K) \leq c \sqrt n L_{K}$ , where $L_{K}$ is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that ${| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \leq C L_{K}$ 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 $m a x_{θ \in S^{n - 1}} {| | ⟨ \cdot, θ ⟩ | |}_{ψ ₂} \geq c ∜ n L_{K}$ . 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 $r ≃ \sqrt n L_{K}$ .

Publié le : 2005-01-01

Zbl 1078.52501

EUDML-ID : urn:eudml:doc:285151

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     title = {On the ps2-behaviour of linear functionals on isotropic convex bodies},
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G. Paouris. On the ψ₂-behaviour of linear functionals on isotropic convex bodies. Studia Mathematica, Tome 166 (2005) pp. 285-299. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-3-7/