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

Let K K be an isotropic convex body in R n R^n . Given ε > 0 ε >0 , how many independent points X i X_i uniformly distributed on K K are needed for the empirical covariance matrix to approximate the identity up to ε ε with overwhelming probability? Our paper answers this question posed by Kannan, Lovász, and Simonovits. More precisely, let X ∈ R n X∈ 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 X is a random point in an isotropic convex body. We show that for any ε > 0 ε >0 , there exists C ( ε ) > 0 C(ε )>0 , such that if N ∼ C ( ε ) n N~ C(ε )\, n and ( X i ) i ≤ N (X_i)i≤ N