  To any $n$-dimensional random vector $X$ we may associate its $L_p$-centroid
body ${\cal Z}_p(X)$ and the corresponding norm. We formulate a conjecture
concerning the bound on the ${\cal Z}_p(X)$-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.
