The Grothendieck constant is strictly smaller than Krivine's bound

Abstract

The classical Grothendieck constant, denoted K G, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing (Equation Presented), a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that K G ≤ π/2 log(1+√2) and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that K G < π/2 log(1+√2)-ε 0 for an explicit constant ε 0 > 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. © 2011 IEEE.