Special orthogonal splittings of L₁²ᵏ

Original · EN

We show that for each positive integer k there is a k× k matrix B with ± 1 entries such that putting E to be the span of the rows of the k× 2k matrix [√kIₖ,B], then E,E is a Kashin splitting: The L₁²ᵏ and the L₂²ᵏ are universally equivalent on both E and E. Moreover, the probability that a random ± 1 matrix satisfies the above is exponentially close to 1.