On the geometry of projective tensor products

Ohad Giladi, Joscha Prochno, Carsten Schütt, Nicole Tomczak-Jaegermann, Elisabeth Werner

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12 Scopus citations

Abstract

In this work, we study the volume ratio of the projective tensor products ℓpnπqnπrn with 1≤p≤q≤r≤∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kašin type whenever 1≤p≤q≤r≤2 or 1≤p≤2≤r≤∞ and q=2. Also, from the Bourgain–Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products ℓp1nπ…⊗πpkn with k∈N and 1≤p1≤…≤pk≤∞.

Original languageEnglish (US)
Pages (from-to)471-495
Number of pages25
JournalJournal of Functional Analysis
Volume273
Issue number2
DOIs
StatePublished - Jul 15 2017

Bibliographical note

Funding Information:
E.W. was supported by NSF grant DMS-1504701.

Keywords

  • Cotype
  • Tensor product
  • Volume ratio

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