On the geometry of projective tensor products

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