TITLE: volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
QUESTION [10 upvotes]: We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

REPLY [5 votes]: You might want to have a look at the work "On the volume of unit balls in Banach spaces" by C. Schuett (Lemma 3.2). He computes
$vol_n(B_{\ell_p^n\otimes_{\varepsilon}\ell_q^n})^{1/n^2}$ up to constants depending only on $p$ and $q$.
In your case, the estimate is
$$vol_n(B_{\ell_1^n\otimes_{\varepsilon}\ell_1^n})^{1/n^2} \asymp n^{-3/2}.$$