Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Understand

Let $K$ be an isotropic convex body in $\R^n$.

• Given $\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits.
• More precisely, let $X\in\R^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix.
• An example of such a vector $X$ is a random point in an isotropic convex body.