Quotient of subspace theorem

The quotient of subspace theorem is an important property of finite dimensional normed spaces, discovered by Vitali Milman.

Let $(X, \| \cdot \|)$ be an $N$ -dimensional normed space. There exist subspaces $Z \subset Y \subset X$ such that the following holds:

The quotient space $E = Y / Z$ is of dimension $\text{dim} E \geq c N$ , where $c > 0$ is a universal constant.
The induced norm $\| \cdot \|$ on $E$ , defined by $\| e \| = \min_{y \in e} \| y \|$ for $e \in E$ , is isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") $Q$ on $E$ , such that

$\frac{\sqrt{Q(e)}}{K} \leq \| e \| \leq K \sqrt{Q(e)}$ for $e \in E,$

with

K > 1

a universal constant.

In fact, the constant $c$ can be made arbitrarily close to 1, at the expense of the constant $K$ becoming large. The original proof allowed

$c(K) \approx 1 - \text{const} / \log \log K$ ;

see references for improved estimates.

[edit] References

V.D.Milman, Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space, Israel seminar on geometrical aspects of functional analysis (1983/84), X, 8 pp., Tel Aviv Univ., Tel Aviv, 1984.

Y. Gordon, On Milman's inequality and random subspaces which escape through a mesh in $R\sp n$, Geometric aspects of functional analysis (1986/87), 84--106, Lecture Notes in Math., 1317, Springer, Berlin, 1988.

G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. xvi+250 pp.