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, 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 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).
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 quadratic form Q on E such that the corresponding Euclidean norm
on E satisfies:
In terms of the multiplicative Banach-Mazur distance d the theorem's conclusion can be formulated as:
where 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.
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