  It is shown that for every $k\in \N$ there exists a Borel probability measure
$\mu$ on $\{-1,1\}^{\R^{k}}\times \{-1,1\}^{\R^{k}}$ such that for every
$m,n\in \N$ and $x_1,..., x_m,y_1,...,y_n\in S^{m+n-1}$ there exist
$x_1',...,x_m',y_1',...,y_n'\in S^{m+n-1}$ such that if $G:\R^{m+n}\to \R^k$ 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')d\mu(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 $K_G$.

Functional Analysis