Krivine Schemes Are Optimal

It is shown that for every k ∈ N there exists a Borel probability measure μ on {−1, 1}Rk × {−1, 1}Rk such that for every m,n ∈ N and x1, . . . , xm, y1, . . . , yn ∈ Sm+n−1 there exist x1, . . . , x ′ m, y ′ 1, . . . , y ′ n ∈ Sm+n−1 such that if G : R → R is a random k× (m+ n) matrix whose entries are i.i.d. standard Gaussian random variables then for all (i, j) ∈ {1, . . . ,m} × {1, . . . , n} we have EG [∫ {−1,1}R×{−1,1}R f(Gxi)g(Gy ′ j)dμ(f, g) ] = 〈xi, yj〉 (1 + C/k)KG , where KG is the real Grothendieck constant and C ∈ (0,∞) is a universal constant. This establishes that Krivine’s rounding method yields an arbitrarily good approximation of KG.