Dvoretzky's theorem

For every natural number k ∈ N and every ε > 0 there exists a natural number N(k, ε) ∈ N such that if (X, ‖·‖) is any normed space of dimension N(k, ε), there exists a subspace E ⊂ X of dimension k and a positive definite quadratic form Q on E such that the corresponding Euclidean norm :| \cdot | = \sqrt{Q(\cdot)} on E satisfies: : |x| \leq \|x\| \leq (1+\varepsilon)|x| \quad \text{for every} \ x \in E. In terms of the multiplicative Banach-Mazur distance d the theorem's conclusion can be formulated as: :d(E,\ \ell_k^2)\leq 1+\varepsilon where \ell_k^2 denotes the standard k-dimensional Euclidean space. Since the unit ball of every normed vector space is a bounded, symmetric, convex set and the unit ball of every Euclidean space is an ellipsoid, the theorem may also be formulated as a statement about ellipsoid sections of convex sets. As a consequence, we have the following statement. For any \epsilon > 0, we call a \epsilon-sphere a convex body K such that there exists a ball B, such that B \subset K \subset (1 + \epsilon) B. Then, for any integer n and any \epsilon \in (0, 1), for all large enough N, and any N-dimensional centrally symmetric body, there exists an n-dimensional subspace V \subset \R^N, such that K \cap V is an \epsilon-sphere. ==Further developments==

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,\varepsilon)\leq\exp(C(\varepsilon)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 : |x| \leq \|x\| \leq (1+\varepsilon)|x| \quad \text{for every} \ x \in E. 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: • there exists such a subspace E with :: k = \dim E \geq C(\varepsilon) \, \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 will be at most :: c_1 \sqrt{\frac{\log N}{N}}. 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, who showed that k*(X) ≥ c2 ε2 log N. Another proof of this result was given by Gideon Schechtman. 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) that is close either to ℓ or to ℓ. Important related results were proved by Tadeusz Figiel, Joram Lindenstrauss and Milman. ==References==