volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:10:42Z http://mathoverflow.net/feeds/question/60062 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60062/volume-of-the-unit-ball-of-the-banach-space-ell-1n-otimes-epsilon-ell-1n volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$? BigBill 2011-03-30T12:53:59Z 2011-03-31T11:50:41Z <p>We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.</p> <p>Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?</p> http://mathoverflow.net/questions/60062/volume-of-the-unit-ball-of-the-banach-space-ell-1n-otimes-epsilon-ell-1n/60175#60175 Answer by Guillaume Aubrun for volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$? Guillaume Aubrun 2011-03-31T11:50:41Z 2011-03-31T11:50:41Z <p>Here is an argument that is certainly overkill and introduces a logarithmic factor which is probably unnecessary.</p> <p>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 </p> <p>$$w(K) = \int_{S} \|x\|_{K^\circ} d\sigma(x)$$</p> <p>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</p> <p>$$w(K^\circ)^{-1} \leq vrad(K^\circ)^{-1} \lesssim vrad(K) \leq w(K)$$</p> <p>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.</p> <p>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).</p> <p>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</p> <p>$$w(K^\circ) \approx \sqrt{n}$$</p> <p>and therefore</p> <p>$$1/\sqrt{n} \lesssim vrad(K) \lesssim \log n/\sqrt{n} .$$ </p> <p>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".</p>