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
^m
Σ
_j=1
ⁿ
a
_ij
ε
_i
δ
_j
: {ε
_i
}
_i=1
^m
, {δ
_j
}
_j=1
ⁿ
⊆{-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 ℝ
²
in order to round the projected vectors, beat the random hyperplane technique, contrary to Krivine's long-standing conjecture.