Abstract
The paper studies the sampling discretization problem for integral norms on subspaces of Lp(μ). Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Guédon and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of Lp[0,1] into ℓpm. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.
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