Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

Let $K$ be an isotropic convex body in $\mathbb{R}^n$. Given $\varepsilon>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\varepsilon$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovász, and Simonovits. More precisely, let $X\in\mathbb{R}^n$ 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. We show that for any $\varepsilon>0$, there exists $C(\varepsilon)>0$, such that if $N\sim C(\varepsilon) n$ and $(X_i)_{i\le N}$ are i.i.d. copies of $X$, then $\Big\Vert\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \operatorname{Id}\Big\Vert \le \varepsilon,$ with probability larger than $1-\exp(-c\sqrt n)$.