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

Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?


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".

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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 $$vol_n(B_{\ell_1^n\otimes_{\varepsilon}\ell_1^n})^{1/n^2} \asymp n^{-3/2}.$$

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