On the embedding of 2-concave Orlicz spaces into $L^1$

Abstract

In [K--S 1] it was shown that $$ \underset π \to {\text{Ave}} (\sum_{i=1}^{n}|x_i a_{π(i)}|^2)^{\frac {1}{2}} $$ is equivalent to an Orlicz norm whose Orlicz function is 2-concave. Here we give a formula for the sequence $a_1, a_2,....,a_n$ so that the above expression is equivalent to a given Orlicz norm.