Abstract
It is shown that for every $k\in \N$ there exists a Borel probability measure
μ
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
∈
S
m+n−1
there exist
x
′
1
,...,
x
′
m
,
y
′
1
,...,
y
′
n
∈
S
m+n−1
such that if $G:\R^{m+n}\to \R^k$ 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 \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∈(0,∞)
is a universal constant. This establishes that Krivine's rounding method yields an arbitrarily good approximation of
K
G
.