John's theorem states that if the Euclidean unit ball is the largest volume ellipsoid in a convex body $K$ in $\mathbb R^d$, then there is a set of unit vectors $u_1,\ldots,u_m$ on the boundary of $K$ such that the identity operator $I$ on $\mathbb R^d$ is a positive linear combination of the diads $u_i\otimes u_i$. Put in another way, I is the expectation of a probability distribution on the set of $n\times n$ real matrices supported on a certain set of rank one matrices.
Motivated by geometric applications, it is natural to ask if the average of few of these random matrices is close to $I$. Our main interest is whether the known positive answer to this question extends from diads to larger classes of matrices.
Joint work with Grigoriy Ivanov and Alexander Polyanskii.