Sampling convex bodies: a random matrix approach

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