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 $x_1,\ldots , x_m,y_1,\ldots ,y_n\in \mathbb {S}^{m+n-1}$ there exist $x_1’,\ldots ,x_m’,y_1’,\ldots ,y_n’\in \mathbb {S}^{m+n-1}$ 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 $(i,j)\in \{1,\ldots ,m\}\times \{1,\ldots ,n\}$ we have \begin{equation*} \mathbb {E}_G\left [\int _{\{-1,1\}^{\mathbb {R}^{k}}\times \{-1,1\}^{\mathbb {R}^{k}}}f(Gx_i’)g(Gy_j’)d\mu (f,g)\right ]=\frac {\langle x_i,y_j\rangle }{(1+C/k)K_G}, \end{equation*} 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 $K_G$.