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 $ext\{dim\}\; E\; geq\; c\; N$, 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 theconstant $K$ becoming large. The original proof allowed

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

see references for improved estimates.

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 $Rsp n$", 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.