Second derivative test for isometric embeddings in $L_p$

Alexander Koldobsky

Second derivative test for isometric embeddings in $L_p$

Functional Analysis 2008-02-03 v1

Authors: Alexander Koldobsky

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.

Keywords

orlicz spaces lp spaces and inequalities metric embeddings

Cite

@article{arxiv.math/9702211,
  title  = {Second derivative test for isometric embeddings in $L_p$},
  author = {Alexander Koldobsky},
  journal= {arXiv preprint arXiv:math/9702211},
  year   = {2008}
}

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