On the thin-shell conjecture for the Schatten classes

TL;DR

This paper advances the understanding of the thin-shell conjecture for Schatten classes by proving it for the operator norm, improving bounds for other cases, and identifying a key negative correlation property necessary for the conjecture.

Contribution

The paper proves the thin-shell conjecture for the operator norm in Schatten classes and improves bounds for other cases, also linking the conjecture to a negative correlation property.

Findings

Confirmed the conjecture for the operator norm.

Improved bounds for Schatten classes beyond the operator norm.

Identified a negative correlation property as necessary for the conjecture.

Abstract

We study the thin-shell conjecture for the Schatten classes. In particular, we establish the conjecture for the operator norm; we also improve on the best known bound for the Schatten classes, due to Barthe and Cordero-Erausquin [F. Barthe and D. Cordero-Erausquin, "Invariances in variance estimates", Proceedings of the London Mathematical Society 106 (2013): 33-64], for a few more cases. Moreover, we show that a necessary condition for the conjecture to be true for any of the Schatten classes is a rather strong negative correlation property: this implies of course that, for the cases for which we already have the conjecture (as for example for the operator norm), but in fact also for all the cases for which we can get a better estimate than the one in [F. Barthe and D. Cordero-Erausquin, "Invariances in variance estimates"], this negative correlation property follows. For the proofs, our starting point is techniques that were employed for the Schatten classes with regard to other problems in [H. K\"onig, M. Meyer and A. Pajor, "The isotropy constants of the Schatten classes are bounded", Mathematische Annalen 312 (1998): 773-783] and [O. Gu\'edon and G. Paouris, "Concentration of mass on the Schatten classes", Annales de l'Institut Henri Poincar\'e, Probabilit\'e et Statistiques 43 (2007): 87-99].