Isotropic Constants of Schatten Class Spaces

Summary

We study the value of the isotropic constant of the unit ball in the Schatten class spaces We prove that, for 2 ≤ p ≤ ∞, this value is bounded by a fixed constant, whereas for 1 ≤ p < 2 it is bounded by c (log n)^1/p-l/2, where c is a fixed constant.

The isotropic constant of a convex symmetric body K ⊂ ℝ^d is a highly important quantity. One of its equivalent definitions is where IKI stands for the volume of K and ||·||₂ is the usual Euclidean norm. See [MP] for other formulations and a full discussion. Estimation of L_K is one of the central problems on the border between local theory and convexity. Bourgain [Bo] has shown that L_K ≲ d^{1/ 4} log d for any K ⊂ℝ^d, where by A ≲ B we mean that A ≲ cB for some universal constant c. He also raised the problem of universal boundedness of the isotropic constant independent of the dimension.

Till now there was no improvement of Bourgain's result in the general case, but boundedness of the isotropic constant was established for some families of bodies (see [MP,Ba,Jl,J2], for example), covering the unit balls of most of the classical spaces.

The case of the Schatten class spaces however, was left out. In [D] we showed that. Here we extend the result to all values of p using a simpler argument. First we shall recall the definiton of the Schatten class spaces.