On the geometry of projective tensor products
| Autor: | Nicole Tomczak-Jaegermann, Carsten Schütt, Elisabeth M. Werner, Joscha Prochno, Ohad Giladi |
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| Rok vydání: | 2017 |
| Předmět: |
Mathematics::Functional Analysis
010102 general mathematics 46A32 46B28 46B07 Banach space Type (model theory) 01 natural sciences Functional Analysis (math.FA) Mathematics - Functional Analysis Combinatorics Tensor product 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Functional Analysis. 273:471-495 |
| ISSN: | 0022-1236 |
| DOI: | 10.1016/j.jfa.2017.03.019 |
| Popis: | In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_{\pi}\ell_q^n\otimes_{\pi}\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. 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 Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ 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 $\ell_{p_1}^n\otimes_{\pi}\dots \otimes_{\pi}\ell_{p_k}^n$ with $k\in\mathbb N$ and $1\leq p_1 \leq \dots\leq p_k \leq \infty$. Comment: 21 pages |
| Databáze: | OpenAIRE |
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