The thin-shell conjecture is a question from the theory of isotropic convex bodies which asks whether the variance of the Euclidean norm, with respect to the uniform measure on an isotropic convex body, can be bounded from above by an absolute constant times the mean of the Euclidean norm (if the answer to this is affirmative, then we have as a consequence that most of the mass of the isotropic convex body is concentrated in an annulus with very small width, a "thin shell''). So far all the general bounds we know depend on the dimension of the bodies, however for certain families of convex bodies, like the $\ell_p$ balls, the conjecture has been resolved optimally. On the other hand, one family of bodies for which the conjecture has been open is the unit balls of the Schatten classes, namely the spaces of square matrices with real, complex or quaternion entries equipped with the $\ell_p$-norm of their singular values.
In this talk we will present two main results concerning the thin-shell conjecture for the Schatten classes. The first result entails the truth of the conjecture for the operator norm (case of $p = \infty$), as well as an improved estimate, compared to the best previously known bound for the Schatten classes due to Barthe and Cordero-Erausquin, for a few more cases. The second result is that a necessary condition for the conjecture to be true for any of the Schatten classes is a rather strong negative correlation property.
This is joint work with Jordan Radke.