Krivine schemes are optimal

TL;DR

The paper proves that Krivine's rounding method can approximate the Grothendieck constant arbitrarily well, using a probabilistic measure and geometric constructions.

Contribution

It establishes the optimality of Krivine schemes in approximating the Grothendieck constant through a probabilistic and geometric approach.

Findings

Existence of a Borel probability measure with specific properties.

Krivine's rounding method achieves arbitrarily close approximation of the Grothendieck constant.

Quantitative bounds involving Gaussian matrices and inner products.

Abstract

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 $K_G$.