On ${\mathcal{Z}}_p$-norms of random vectors

To any $n$-dimensional random vector $X$ we may associate its $L_p$-centroid body ${\mathcal{Z}}_p(X)$ and the corresponding norm. We formulate a conjecture concerning the bound on the ${\mathcal{Z}}_p(X)$-norm of $X$ and show that it holds under some additional symmetry assumptions. We also relate our conjecture with estimates of covering numbers and Sudakov-type minorization bounds.