Dvoretzky's theorem
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s,[1] answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids.
A new proof found by Vitali Milman in the 1970s[2] was one of the starting points for the development of asymptotic geometric analysis (also called asymptotic functional analysis or the local theory of Banach spaces).[3]
In 1971, Vitali Milman gave a new proof of Dvoretzky's theorem, making use of the concentration of measure on the sphere to show that a random k-dimensional subspace satisfies the above inequality with probability very close to 1. The proof gives the sharp dependence on k:
where the constant C(ε) only depends on ε.
We can thus state: for every ε > 0 there exists a constant C(ε) > 0 such that for every normed space (X, ‖·‖) of dimension N, there exists a subspace E ⊂ X of dimension
k ≥ C(ε) log N and a Euclidean norm |⋅| on E such that
More precisely, let SN − 1 denote the unit sphere with respect to some Euclidean structure Q on X, and let σ be the invariant probability measure on SN − 1. Then:
Here c1 is a universal constant. For given X and ε, the largest possible k is denoted k*(X) and called the Dvoretzky dimension of X.
The dependence on ε was studied by Yehoram Gordon,[4][5] who showed that k*(X) ≥ c2 ε2 log N. Another proof of this result was given by Gideon Schechtman.[6]
Noga Alon and Vitali Milman showed that the logarithmic bound on the dimension of the subspace in Dvoretzky's theorem can be significantly improved, if one is willing to accept a subspace that is close either to a Euclidean space or to a Chebyshev space. Specifically, for some constant c, every n-dimensional space has a subspace of dimension k ≥ exp(c√log N) that is close either to ℓk
2 or to ℓk
∞.[7]
Important related results were proved by Tadeusz Figiel, Joram Lindenstrauss and Milman.[8]