Krivine schemes are optimal

Abstract

It is shown that for every $k\in \mathbb{N}$ there exists a Borel probability measure $\mu$ on $\{-1,1{\}}^{{\mathbb{R}}^k}\times \{-1,1{\}}^{{\mathbb{R}}^k}$ such that for every $m,n\in \mathbb{N}$ and $x_1,...,x_m,y_1,...,y_n\in S^{m+n-1}$ there exist $x_1^{\prime },...,x_m^{\prime },y_1^{\prime },...,y_n^{\prime }\in S^{m+n-1}$ such that if $G:{\mathbb{R}}^{m+n}\to {\mathbb{R}}^k$ is a random $k\times (m+n)$ matrix whose entries are i.i.d. standard Gaussian random variables then for all $(i,j)\in 1,...,m\times 1,...,n$ we have \E_G[\int_{{-1,1}^{\R^{k}}\times {-1,1}^{\R^{k}}}f(Gx_i')g(Gy_j')dμ(f,g)]=\frac{<x_i,y_j>}{(1+C/k)K_G}, where $K_G$ is the real Grothendieck constant and $C\in (0,\infty )$ is a universal constant. This establishes that Krivine's rounding method yields an arbitrarily good approximation of $K_G$.