  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 $1\leq p_1 \leq \dots\leq p_k \leq \infty$.
