Timeline for Intuitive explanation of Dvoretzky's theorem
Current License: CC BY-SA 3.0
10 events
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| Sep 30, 2013 at 12:33 | comment | added | BS. | @Mark and Igor : the half dimension of spherical section is the clue ! Let the ellipsoid be $\sum_1^{2k} a_i x_i^2=1$ and $(a_i)_{i=1..2k}$ be increasing. Take $c$ in $[a_k,a_{k+1}]$. Then the subspace $x_{k+1-i}=t_ix_i$, $i=1..k$ will do the job if $a_i+t_i^2 a_{k+1-i}=(1+t_i^2) c$, which is possible by choice of $c$ (maybe with infinite $t_i$). An obvious adaptation works in the odd $2k+1$ dimensional case, giving a $k+1$-dim spherical section. | |
| Sep 30, 2013 at 12:06 | comment | added | Mark Meckes | Milman's proof shows that after applying a linear transformation, most low-dimensional sections have an almost spherical shape. Applying the inverse transformation, you get almost ellipsoidal sections, but the "most" part gets lost. But, since ellipsoids have (about half-dimensional) precisely spherical sections, you still get almost spherical sections of the original body. I'm sure I've seen a short proof of the existence of spherical sections of ellipsoids in some lecture notes by Schechtman, but I can't find it now. | |
| Sep 30, 2013 at 11:33 | comment | added | Igor Rivin | @BS yes, I agree, that may be more tractable, intuitively... | |
| Sep 30, 2013 at 9:06 | comment | added | alvarezpaiva | @Igor: the usual formulation is in terms of the Banach-Mazur distance which doesn't see a difference between ellipsoids and balls. | |
| Sep 30, 2013 at 7:06 | comment | added | BS. | @IgorRivin : perhaps you should ask why high dimensional ellipsoids have large almost spherical sections, intuitively ? | |
| Sep 29, 2013 at 21:20 | comment | added | Igor Rivin | That's why Dvoretzky's theorem is so cool! The wikipedia has a correct statement: en.wikipedia.org/wiki/Dvoretzky's_theorem | |
| Sep 29, 2013 at 21:14 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| Sep 29, 2013 at 21:13 | comment | added | Carlo Beenakker | @IgorRivin --- my very limited understanding of Dvoretzky/Milman is based on Keith Ball's "Elementary Introduction to Modern Convex Geometry", where it is shown that "any symmetric convex body in $R^n$ has almost ellipsoidal sections of dimension about $\log n$." I would be surprised if almost all sections would be spherical, ellipsoidal seems the generic thing to expect. | |
| Sep 29, 2013 at 20:46 | comment | added | Igor Rivin | I am a little confused -- low-dimensional sections have an almost SPHERICAL shape (and Milman, as you say, makes this into a "most sections" statement). Also, in the paper you mention they seem to use Dvoretzky's theorem (rather than give any sort of argument for it). Am I confused? | |
| Sep 29, 2013 at 20:41 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |