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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.30039 |
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| _version_ | 1866914435556179968 |
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| author | Jones, Chris Malavolta, Giulio |
| author_facet | Jones, Chris Malavolta, Giulio |
| contents | 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 $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_30039 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound Jones, Chris Malavolta, Giulio Functional Analysis Quantum Physics 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 $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator. |
| title | The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound |
| topic | Functional Analysis Quantum Physics |
| url | https://arxiv.org/abs/2603.30039 |