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

• Autorzy: G. Paouris
• Języki publikacji: EN

Abstrakty

EN

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 $r ≃ √n L_{K}$.

Kategorie tematyczne

• 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
• 52A40: Inequalities and extremum problems
• 52A20: Convex sets in n dimensions (including convex hypersurfaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 168
• Numer: 3
• Strony: 285-299

Daty

wydano

2005

Twórcy

autor

G. Paouris

• Department of Mathematics, University of Crete, Iraklion 714-09, Crete, Greece

Identyfikatory

DOI

10.4064/sm168-3-7