The problem
Theorem: for every ε > 0 and integer k, there exists N(k,ε) such that every normed space of dimension n ≥ N contains a k-dimensional subspace E with the Banach–Mazur distance d(E, ℓ²_k) ≤ 1 + ε.
The Dvoretzky Theorem (Low-Dimensional Sections of High-Dimensional Balls)
historicReferences
History & significance
Emerged from the Banach-era question of whether infinite-dimensional spaces must contain infinite-dimensional Euclidean subspaces (they need not, as James showed). Dvoretsky proved the finite-dimensional affirmative in 1960 via spherical sections of the unit ball. Milman's 1971 concentration-of-measure proof reduced the dimension bound to exponential in 1/ε² — the birth of concentration phenomena as a field. Figiel–Lindenstrauss–Milnor showed k ≤ c log n is best possible.
The resolution (human proof)
Proven by Aryeh Dvoretsky, 1959-60, with the quantitative refinement by Vitali Milman (1971) introducing concentration of measure to Banach space theory — arguably creating asymptotic geometric analysis as a discipline. Modern proofs use cotype-2 constants and the quotient of subspace theorem (Milman). The theorem explains why high-dimensional data looks locally flat, connecting to compressed sensing and random matrix theory.