 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