Primal/Dual Method for the Grothendieck Constant

Abstract

We determine the hundredths digit of the real Grothendieck constant by proving $1.773\le K_G\le 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G\approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bounds are obtained by a heuristic primal/dual search for these objects, followed by rigorous {\em certification} of their values. The search and certification are performed using AI tools, and in particular the rigorous certificates are very large (analytical reductions to thousands of numerical inequalities).