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,[1] 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[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]

  1. ^ Dvoretzky, A. (1961). "Some results on convex bodies and Banach spaces". Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960). Jerusalem: Jerusalem Academic Press. pp. 123–160.
  2. ^ Milman, V. D. (1971). "A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies". Funkcional. Anal. I Prilozhen. (in Russian). 5 (4): 28–37.
  3. ^ Gowers, W. T. (2000). "The two cultures of mathematics". Mathematics: frontiers and perspectives. Providence, RI: Amer. Math. Soc. pp. 65–78. ISBN 978-0-8218-2070-4. The full significance of measure concentration was first realized by Vitali Milman in his revolutionary proof [Mil1971] of the theorem of Dvoretzky ... Dvoretzky's theorem, especially as proved by Milman, is a milestone in the local (that is, finite-dimensional) theory of Banach spaces. While I feel sorry for a mathematician who cannot see its intrinsic appeal, this appeal on its own does not explain the enormous influence that the proof has had, well beyond Banach space theory, as a result of planting the idea of measure concentration in the minds of many mathematicians. Huge numbers of papers have now been published exploiting this idea or giving new techniques for showing that it holds.