We denote by $\otimes\_{\epsilon}$ the injective Banach tensor product.


Which is the asymptotic volume of the unit ball of the Banach space $\ell\_1^n\otimes\_{\epsilon}\ell\_1^n$?



volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
Here is an argument that is certainly overkill and introduces a logarithmic factor which is probably unnecessary.


Let $K$ be the unit ball in $\ell\_1^n \otimes\_\epsilon \ell\_1^n$ and $K^\circ$ the polar body (the unit ball in $\ell\_{\infty}^n \otimes\_\pi \ell\_{\infty}^n$). It is convenient to introduce the volume radius $vrad(K)=(vol(K)/vol(B\_2^n))^{1/n}$ (this is the radius of the Euclidean ball with the same volume as $K$) and the mean (half-)width 


$$ w(K) = \int\_{S} \|x\|\_{K^\circ} d\sigma(x)$$


where $\sigma$ is the uniform probability measure on the Euclidean sphere $S$ in $\mathbf{R}^n \otimes \mathbf{R}^n$. One has the following chain of inequalities


$$ w(K^\circ)^{-1} \leq vrad(K^\circ)^{-1} \lesssim vrad(K) \leq w(K) $$


The first and third inequalities are Uryshon's inequality and the central one is the reverse Santalo inequality (a deep theorem). Now there is another deep theorem that whenever a $n$-dimensional symmetric convex body is in $\ell$-position (which means that in some sense it is well-balanced) the product $w(K)w(K^\circ)$ is bounded by $C \log n$, and thefore the four quantities in the chain of inequalities above are comparable up to a logarithmic factor.


A convex body is in $\ell$-position as long as it has "enough symmetries" (i.e. the group
of isometries acts irreducibly, this is the case here).


The simplest quantity to estimate seems to be $w(K^\circ)$. Replacing spherical integration by Gaussian integration, one essentially has to compute the norm of a Gaussian matrix as an operator from $\ell\_{\infty}^n$ to $\ell\_1^n$. If I am correct one obtains


$$ w(K^\circ) \approx \sqrt{n} $$


and therefore


$$ 1/\sqrt{n} \lesssim vrad(K) \lesssim \log n/\sqrt{n} .$$ 


Depending on your background (who are you ??), this may be quite obscure to you. I think everything relevant here is in the book by Gilles Pisier "the volume of convex bodies and Banach space geometry".


