Dvoretzky's theorem
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, Dvoretzky's theorem is an important structural theorem about
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s proved by Aryeh Dvoretzky in the early 1960s, answering a question of
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
. 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 In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
has low-dimensional sections that are approximately
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s. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called ''asymptotic
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
'' or the ''local theory of Banach spaces'').


Original formulations

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 In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
''Q'' on ''E'' such that the corresponding Euclidean norm :, \cdot , = \sqrt on ''E'' satisfies: : , x, \leq \, x\, \leq (1+\varepsilon), x, \quad \text \ 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 Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
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 \ x \in E. More precisely, let ''S''''N'' − 1 denote the unit sphere with respect to some Euclidean structure ''Q'' on ''X'', and let ''σ'' be the invariant probability measure on ''S''''N'' − 1. Then: * there exists such a subspace ''E'' with :: k = \dim E \geq C(\varepsilon) \, \left(\frac\right)^2 \, N. * For any ''X'' one may choose ''Q'' so that the term in the brackets will be at most :: c_1 \sqrt. Here ''c''1 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'') ≥ ''c''2 ''ε''2 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., expanded in "The dimension of almost spherical sections of convex bodies", Acta Math. 139 (1977), 53–94.


References


Further reading

* {{Functional analysis Banach spaces Asymptotic geometric analysis Theorems in functional analysis