Abstract
In this work, we study the volume ratio of the projective tensor products ℓpn⊗πℓqn⊗πℓrn 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 ℓp1n⊗π…⊗πℓpkn with k∈N and 1≤p1≤…≤pk≤∞.
Original language | English (US) |
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Pages (from-to) | 471-495 |
Number of pages | 25 |
Journal | Journal of Functional Analysis |
Volume | 273 |
Issue number | 2 |
DOIs | |
State | Published - Jul 15 2017 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2017 Elsevier Inc.
Keywords
- Cotype
- Tensor product
- Volume ratio