  For a random vector X in R^n, we obtain bounds on the size of a sample, for
which the empirical p-th moments of linear functionals are close to the exact
ones uniformly on an n-dimensional convex body K. 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 L_p. We also prove that for an isotropic
log-concave random vector, we only need about n^{p/2} \log n sample points so
that the empirical p-th 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.
