ψ ₂-Estimate for the Euclidean Norm on a Convex Body in Isotropic Position

Abstract

Let ℝⁿ be the n-dimensional euclidean space with fixed scalar product (·,·) and the norm |x|² = (x, x). Denote D _n = {x ∈ ℝⁿ | |x| ≤ 1} the unit euclidean ball, |A| = vol _n A the Lebesgue n-dimensional measure. Let K be compact convex body in $\mathbb{R}^n ,b = \frac{1} {{|K|}}\int_K {xdx}$ be its centroid and $M = (m_{ij} )_{i,j = 1}^n$ be the matrix of inertia of K with entries $m_{ij} = \frac{1} {{|K|}}\int_K {x_i x} _j dx.$