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

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.

Functional Analysis (math.FA); Quantum Physics (quant-ph)

arXiv:2603.30039 [math.FA]

(or arXiv:2603.30039v2 [math.FA] for this version)

https://doi.org/10.48550/arXiv.2603.30039 Focus to learn more arXiv-issued DOI via DataCite

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