Ja n 20 03 Special orthogonal splittings of L 2 k 1

We show that for each positive integer k there is a k × k matrix B with ±1 entries such that letting K1 be the symmetric convex hull of the rows of B and K2 the symmetric convex hull of √ k times the canonical unit vector basis of Rk (= √ kBk 1 ), then K1∩K2 lies between two universal multiples of the Euclidean unit ball, Bk 2 . Moreover, the probability that a random ±1 matrix satisfies the above is exponentially close to 1. It follows that, putting E to be the span of the rows of the k×2k matrix [ √ kIk, B], then, with high probability over k × k matrices B with independent ±1 entries, E,E⊥ is a Kashin splitting: The L2k 1 and the L 2k 2 are universally equivalent on both E and E⊥.