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

G. Paouris. "On the ψ₂-behaviour of linear functionals on isotropic convex bodies." Studia Mathematica 168.3 (2005): 285-299. <http://eudml.org/doc/285151>.

@article{G2005,
abstract = {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 $r ≃ √n L_\{K\}$.},
author = {G. Paouris},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {285-299},
title = {On the ψ₂-behaviour of linear functionals on isotropic convex bodies},
url = {http://eudml.org/doc/285151},
volume = {168},
year = {2005},
}

TY - JOUR
AU - G. Paouris
TI - On the ψ₂-behaviour of linear functionals on isotropic convex bodies
JO - Studia Mathematica
PY - 2005
VL - 168
IS - 3
SP - 285
EP - 299
AB - 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 $r ≃ √n L_{K}$.
LA - eng
UR - http://eudml.org/doc/285151
ER -