Timeline for John-type theorems: trading structure for accuracy?
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6 events
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| Sep 19, 2023 at 11:26 | comment | added | Guillaume Aubrun | Since you mention zonoids, here is an approximation result for them: any n -dimensional zonoid can be approximated with Banch-Mazur distance ⩽1+ϵ by the Minkowski sum of N=O(nlogn/ϵ2) segments. See M. Talagrand, Embedding subspaces of L_1 into l_1^N, Proc. AMS (1990), cf jstor.org/stable/2048283 | |
| Sep 14, 2023 at 15:18 | comment | added | Mikael de la Salle | On the other hand, Milman's QS-Theorem gives a very significant improvement on John's theorem if you allow not only slices, but also projections, see en.wikipedia.org/wiki/Quotient_of_subspace_theorem | |
| Sep 14, 2023 at 1:48 | comment | added | Mikael de la Salle | Of course, there quite some room between $\sqrt{d}$ and $\sqrt{d/\log d}$, so this does not really answer your question. | |
| Sep 14, 2023 at 1:38 | comment | added | Mikael de la Salle | See Theorem 3 for $p=2$ in Pisier's Remarques sur un résultat non publié de B. Maurey numdam.org/item/?id=SAF_1980-1981____A5_0 | |
| Sep 14, 2023 at 1:38 | comment | added | Mikael de la Salle | Regarding your second question : Not sure in what kind of improvement you are looking for, but a theorem by Maurey seems to indicate that looking for large-dimensional slices cannot very significantly (more than up to log factors) improve John's theorem: every subspace of $\ell_\infty^n$ of dimension $d$ is at Banach-Mazur distance $\geq C \sqrt{\frac{d}{\log n}}$ from a Euclidean space. In fact, this is even a lower bound on the type 2 constant of the dual, which is an obvious upper bound on the Banach-Mazur distance. | |
| Sep 13, 2023 at 19:55 | history | asked | Terry Tao | CC BY-SA 4.0 |