Sampling convex bodies: a random matrix approach

Abstract

We prove the following result: for any ε > 0 \varepsilon >0 , only C ( ε ) n C(\varepsilon )n sample points are enough to obtain ( 1 + ε ) (1+\varepsilon ) -approximation of the inertia ellipsoid of an unconditional convex body in R n \mathbf {R}^n . Moreover, for any ρ > 1 \rho >1 , already ρ n \rho n sample points give isomorphic approximation of the inertia ellipsoid. The proofs rely on an adaptation of the moments method from Random Matrix Theory.