On the geometry of projective tensor products

Abstract

In this work, we study the volume ratio of the projective tensor products ${\ell }_p^n{\otimes }_{\pi }{\ell }_q^n{\otimes }_{\pi }{\ell }_r^n$ with $1\le p\le q\le r\le \infty$. 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 $1\le p\le q\le r\le 2$ or $1\le p\le 2\le r\le \infty$ 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 ${\ell }_{p_1}^n{\otimes }_{\pi }\cdots {\otimes }_{\pi }{\ell }_{p_k}^n$ with $k\in \mathbb{N}$ and $1\le p_1\le \cdots \le p_k\le \infty$.

21 pages