The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

TL;DR

This paper proves that the Grothendieck constant exceeds the Davie-Reeds lower bound by a small positive margin using perturbative analysis of the associated operator.

Contribution

It demonstrates that the Davie-Reeds bound is not tight, establishing a strictly larger lower bound for the Grothendieck constant through novel perturbation techniques.

Findings

The Grothendieck constant is strictly larger than the Davie-Reeds bound by at least 10^{-12}.

Near-extremizers for the Davie-Reeds problem have significant degree-3 Hermite coefficients.

Small cubic perturbations increase the operator's integrality gap.

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 $K_{G} \ge K_{DR} + 10^{-12}$, 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 $\Omega(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.