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 semidefinite relaxation of the problem of computing max {Σ_i-1^mΣ_j=1ⁿa_ijε_iδ_j: {ε_i}_i=1^m, {δ_j}_j=1ⁿ⊆{-1,1} } a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that KG ≤ 2log (1+√2)/π and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that KG <; 2log (1+√2)/π for an explicit constant ε_o >; 0. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of ℝ² in order to round the projected vectors, beat the random hyperplane technique, contrary to Krivine´s long-standing conjecture.