Abstract
The Grothendieck constant K_G is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of K_G is unknown. The best known lower bound on K_G was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that K_G ≥ K_DR + 10⁻¹², where K_DR denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has Ω(1) weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.