Hadamard products and moments of random vectors

We derive sharp comparison inequalities between weak and strong moments of random vectors in arbitrary finite dimensional Banach space. As an application, we show that the p-summing constant of any finite dimensional Banach space is upper bounded, up to a universal constant, by the p-summing constant of the Hilbert space of the same dimension. We also apply our result to the concentration of measure theory for log-concave random vectors in Euclidean spaces.