Dvoretzky's theorem

From Wikipedia, the free encyclopedia

In mathematics, in the theory of Banach spaces, Dvoretzky's theorem is an important structural theorem proved by Aryeh Dvoretzky in the early 1960s. It answered a question of Alexander Grothendieck. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic functional analysis (also called the local theory of Banach spaces).

[edit] Original formulation

For every k \in \mathbf{N} and every ε > 0 there exists N(k, \epsilon) \in \mathbf{N} such that if (X, \| \cdot \|) is a Banach space of dimension \geq N(k, \epsilon), there exist a subspace E \subset X of dimension k and a positive quadratic form Q on E such that the corresponding Euclidean norm

| \cdot | = \sqrt{Q(\cdot)}

on E satisfies:

|x| \leq \|x\| \leq (1+\epsilon)|x| \quad \text{for every} \quad x \in E.

[edit] 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:

N(k,\epsilon)\leq\exp(C(\epsilon)k).

Equivalently, for every Banach space (X, \| \cdot \|) of dimension N, there exists a subspace E \subset X of dimension k \geq c(\epsilon) \log N and a Euclidean norm | \cdot | on E such that the inequality above holds.

More precisely, let Sn − 1 be the unit sphere with respect to some Euclidean structure Q, and let σ be the invariant probability measure on Sn − 1. Then:

  • There exists such a subspace E with
k = \dim E \geq c(\epsilon) \, \left(\frac{\int_{S^{n-1}} \| \xi \| d\sigma(\xi)}{\max_{\xi \in S^{n-1}} \| \xi \|}\right)^2 \, n.
  • For any X one may choose Q so that the term in the brackets be at most
c_1 \sqrt{\frac{\log N}{N}}.

Here c1 is a universal constant. The best possible k is denoted k_\ast(X) and called the Dvoretzky dimension of X.

The dependence on ε was studied by Yehoram Gordon, who showed that k_\ast(X) \geq c_2 \epsilon^2 \log N .

Important related results were proved by Tadeusz Figiel, Joram Lindenstrauss and Vitali Milman.

[edit] References

  • A.Dvoretzky, Some results on convex bodies and Banach spaces, 1961 Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) pp. 123--160 Jerusalem Academic Press, Jerusalem; Pergamon, Oxford
  • V.D.Milman, A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies (Russian), Funkcional. Anal. i Prilozhen. 5 (1971), no. 4, 28--37
  • T. Figiel, J. Lindenstrauss, J., V.D.Milman, The dimension of almost spherical sections of convex bodies, Bull. Amer. Math. Soc. 82 (1976), no. 4, 575--578.
  • Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), no. 4, 265–289.
  • Y. Gordon, Gaussian processes and almost spherical sections of convex bodies, Ann. Probab. 16 (1988), no. 1, 180--188