Second derivative test for isometric embeddings in $L_p$

Abstract

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.