The Grothendieck Constant is Strictly Smaller than Krivine's Bound

The classical Grothendieck constant, denoted $K_G$, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing$$\max \{\sum_{i=1}^m\sum_{j=1}^n a_{ij} \epsilon_i\delta_j: \{\epsilon_i\}_{i=1}^m,\{\delta_j\}_{j=1}^n\subseteq \{-1,1\}\},$$a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that $K_G\leq \pi / (2\log(1+\sqrt{2}))$ and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that $K_G 0$. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of $R^2$ in order to round the projected vectors, beat the random hyper plane technique, contrary to Krivine's long-standing conjecture.