  An old problem of P. Levy is to characterize those Banach spaces which embed
isometrically in $L_p.$ We show a new criterion in terms of the second
derivative of the norm. As an application, we show that if $M$ is a twice
differentiable Orlicz function with $M'(0)=M''(0)=0$ then the $n$-dimensional
Orlicz space $\ell_M^n,\ n\ge 3,$ does not embed isometrically in $L_p$ with
$0<p\le 2.$ These results generalize and clear up the recent solution to the
1938 Schoenberg's problem on positive definite functions.
