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 $M^{\prime }(0)=M^{\prime \prime }(0)=0$ 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.

References in corpus (1)

• Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes