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,...,x_m, y₁,..., y_n ∈ S^m+n−1 there exist x′₁,..., x′_m, y′₁,..., y′_n ∈ S ^m+n−1 such that if G : ℝ^m+n → ℝ^k 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 where K_G 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 K_G.