Concentration of the distance in finite dimensional normed spaces

We prove that in every finite dimensional normed space, for "most" pairs (x, y) of points in the unit ball, \\x-v\\ is more than root 2(1 - epsilon). As a consequence, we obtain a result proved by Bourgain, using QS-decomposition, that guarantees an exponentially large number of points in the unit ball any two of which are separated by more than root 2(1 - epsilon).