Lp-moments of random vectors via majorizing measures

For a random vector X in Rn, we obtain bounds on the size of a sample, for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body K⊂Rn. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of Lp. We also prove that for an isotropic log-concave random vector, we only need ⌊np/2logn⌋ sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.