Dvoretzky's theorem
In mathematics, in the theory of Banach spaces, Dvoretzky's theorem is an important structural theorem proved by Aryeh Dvoretzky in the early 1960s.[1] It answered a question of Alexander Grothendieck. 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]
Original formulation
For every natural number k ∈ N and every ε > 0 there exists N(k, ε) ∈ N such that if (X, ‖.‖) is a Banach space of dimension N(k, ε), there exist a subspace E ⊂ X of dimension k and a positive quadratic form Q on E such that the corresponding Euclidean norm
on E satisfies:
Further development
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:
Equivalently, for every Banach 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 the inequality above holds.
More precisely, let Sn − 1 denote the unit sphere with respect to some Euclidean structure Q, and let σ be the invariant probability measure on Sn − 1. Then:
- For any Q, there exists such a subspace E with
- For any X one may choose Q so that the term in the brackets will be at most
Here c1 is a universal constant. The best 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]
References
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- ↑ Lua error in package.lua at line 80: module 'strict' not found., expanded in "The dimension of almost spherical sections of convex bodies", Acta Math. 139 (1977), 53–94.