Comments on: The pseudorandom subspace problem http://tcsmath.wordpress.com/2008/05/04/the-pseudorandom-subspace-problem/ some mathematics of theoretical computer science Fri, 16 Oct 2009 11:37:51 +0000 http://wordpress.com/ hourly 1 By: jrluw http://tcsmath.wordpress.com/2008/05/04/the-pseudorandom-subspace-problem/#comment-8 jrluw Thu, 08 May 2008 04:00:42 +0000 http://tcsmath.wordpress.com/?p=35#comment-8 Hi Aravind, Great questions. First, one can actually take $latex A$ to be the <em>median</em> of $latex x \mapsto \|x\|$ as $latex x$ ranges over the Euclidean sphere $latex S^{N-1}$. (This hints at one of the core principles behind Milman's proof: The value of $latex \|\cdot\|$ is sharply concentrated on $latex S^{N-1}$.) In particular, $latex A$ does not depend on $latex \varepsilon$. The extremal space for $latex c(\varepsilon)$ is unknown, in fact there is still a big gap between the upper and lower bounds: $latex \frac{1}{\log \varepsilon^{-1}} \gtrsim c(\varepsilon) \gtrsim \frac{\varepsilon}{(\log \varepsilon^{-1})^2}$. The upper bound comes from considering the $latex \ell_\infty$ norm, and the best lower bound is due to Schechtman. Hi Aravind,

Great questions. First, one can actually take A to be the median of x \mapsto \|x\| as x ranges over the Euclidean sphere S^{N-1}. (This hints at one of the core principles behind Milman’s proof: The value of \|\cdot\| is sharply concentrated on S^{N-1}.) In particular, A does not depend on \varepsilon.

The extremal space for c(\varepsilon) is unknown, in fact there is still a big gap between the upper and lower bounds: \frac{1}{\log \varepsilon^{-1}} \gtrsim c(\varepsilon) \gtrsim \frac{\varepsilon}{(\log \varepsilon^{-1})^2}. The upper bound comes from considering the \ell_\infty norm, and the best lower bound is due to Schechtman.

]]> By: aravind srinivasan http://tcsmath.wordpress.com/2008/05/04/the-pseudorandom-subspace-problem/#comment-7 aravind srinivasan Thu, 08 May 2008 03:27:11 +0000 http://tcsmath.wordpress.com/?p=35#comment-7 hi james, welcome to the ToC blogosphere -- you have started with a bang with such a detailed post! i have two questions about Dvoretzky's Theorem as you have stated it: (i) as stated, your quantification allows A to be a function of N, the norm, and epsilon (as it is just a normalization) -- is it independent of any of these -- perhaps of both N and epsilon? or am i being silly here? (ii) since c(epsilon) is implicitly the infimum over all possible norms -- is it known what the extremal norm is? hi james, welcome to the ToC blogosphere — you have started with a bang with such a detailed post! i have two questions about Dvoretzky’s Theorem as you have stated it: (i) as stated, your quantification allows A to be a function of N, the norm, and epsilon (as it is just a normalization) — is it independent of any of these — perhaps of both N and epsilon? or am i being silly here? (ii) since c(epsilon) is implicitly the infimum over all possible norms — is it known what the extremal norm is?

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