Intuitive explanation of Dvoretzky's theorem I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number of proofs but all of them seem a bit technical... 
 A: To develop some intuition, the following argument might help, suggested (and dismissed) by K. Villaverde, O. Kosheleva, and M. Ceberio, Why Ellipsoid Constraints, Ellipsoid Clusters, and Riemannian Space-Time: Dvoretzky's Theorem Revisited.
A stronger version of Dvoretzky’s theorem (due to Milman) asserts that almost all low-dimensional sections of a convex set have an almost ellipsoidal shape. An $n$-dimensional section consists of points $(x_1,x_2,\ldots x_n)$ such that $g(x_1,x_2,\ldots x_n)\leq 0$. Generically, this function $g$ will be smooth and a Taylor expansion to second order would be a good approximation,
$$\sum_{i,j=1}^n a_{ij}x_i x_j+\sum_{i=1}^n b_i x_i \leq a_0,$$
producing an ellipsoid.
A: There is a more difficult proof than the quantitative finite dimensional proofs that gives only the qualitative version of Dvoretzky's theorem but is arguably more intuitive.  You use Ramsey's theorem to prove that if $X$ is an infinite dimensional Banach space, then there is a Banach space $Y$ that has a monotonely unconditional basis $(e_n)$ s.t. $(e_n)$ is isometrically equivalent to every subsequence of itself and such that $Y$ is finitely representable in $X$ (meaning that for every $\epsilon > 0$, every finite dimensional subspace of $Y$ is $1+\epsilon$-isomorphic to a subspace of $X$).  This is at the beginning of the Brunel-Sucheston spreading model theory and is elementary.  So $Y$ looks a bit like the spaces $\ell_p$, $1\le p < \infty$, and $c_0$.  Now $c_0$ is universal for finite dimensional spaces (up to $1+\epsilon$), and $L_p$ for all $p$ contains $\ell_2$ isometrically (span of IID $N(0,1)$ random variables when $p<\infty$), and $L_p$ is finitely representable in $\ell_p$, so this is a pretty good hint that Dvoretzky's theorem is true.  To finish the proof, just apply Krivine's theorem, which says that for some $1\le p \le \infty$, the space $\ell_p$ is finitely represented in $Y$ (in fact, for each $n$ there are disjointly supported elements in $Y$ that are $1+\epsilon$-equivalent to the unit vector basis of $\ell_p^n$).  Krivine's theorem is proved in the Springer Lecture Notes volume written by V. Milman and S. Schechtman.  
If you are willing to settle for subspaces of $Y$ that are just uniformly isomorphic to $\ell_2^n$, you can replace Krivine's theorem with more elementary arguments.  Tzfriri did that in  
Tzafriri, L. On Banach spaces with unconditional bases. Israel J. Math. 17 (1974), 84–93.
