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arXiv:2603.30039

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

Chris Jones, Giulio Malavolta

math.FAquant-phSubmitted Mar 31, 2026• Announced Apr 1, 2026• Revised Oct 6, 2026 (v2)
v1 · Mar 31, 2026v2 · Oct 6, 2026
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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 $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

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15 pages

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The Grothendieck constant KGK_{G}KG​ 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 KGK_{G}KG​ is unknown. The best known lower bound on KGK_{G}KG​ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that KG≥KDR+10−12K_{G} \ge K_{DR} + 10^{-12}KG​≥KDR​+10−12, where KDRK_{DR}KDR​ 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)Ω(1)Ω(1) weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.
We show that 1.4<KGC<1.4048985547461.4<K_G^{\mathbb{C}}<1.4048985547461.4<KGC​<1.404898554746, where KGCK_G^{\mathbb{C}}KGC​ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of 1.40490913…1.40490913\ldots1.40490913… from 1987, and the lower bound improves on Davie's bound of 1.338071.338071.33807 from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.
Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinite Grothendieck problem -- based on rounding from the standard relaxation -- that achieves a ratio of 2/π+Θ(1/n)2/π+ Θ(1/{\sqrt n})2/π+Θ(1/n​), whereas the previous best is 2/π+Θ(1/n)2/π+ Θ(1/n)2/π+Θ(1/n). We further show a corresponding integrality gap of 2/π+O~(1/n1/3)2/π+\tilde{O}(1/n^{1/3})2/π+O~(1/n1/3).
We propose candidate lower bounds on the real Grothendieck constants KG(d)K_{\mathrm G}(d)KG​(d) and their infinite-dimensional limit KGK_{\mathrm G}KG​ using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply KG≥9π/16≃1.767146K_{\mathrm G}\geq9π/16\simeq1.767146KG​≥9π/16≃1.767146, improving the rigorous lower bound 6π/11≃1.7135966π/11\simeq1.7135966π/11≃1.713596. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to 242424. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on KG(d)K_{\mathrm G}(d)KG​(d) in the studied dimensions from 444 to 242424, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.
Grothendieck constants KG(d)K_G(d)KG​(d) bound the advantage of ddd-dimensional strategies over 111-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from d=2d=2d=2, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For d∈{3,4,5}d\in\{3,4,5\}d∈{3,4,5}, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for d⩽9d\leqslant9d⩽9. For d∈{4,7,8}d\in\{4,7,8\}d∈{4,7,8}, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants KG(d↦2)K_G(d\mapsto2)KG​(d↦2) as the advantage of complex ddd-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on KG(d↦2)K_G(d\mapsto2)KG​(d↦2).
In this paper, we prove that KG(3)<KG(4){K}_{G}(3)<{K}_{G}(4)KG​(3)<KG​(4), where KG(d){K}_{G}(d)KG​(d) denotes the Grothendieck constant of order ddd. To this end, we use a branch-and-bound algorithm commonly used in the solution of NP-hard problems. It has recently been proven that ${K}_{G}(3)\ensuremath{\le}1.4644$. Here we prove that ${K}_{G}(4)\ensuremath{\ge}1.4841$, which has implications for device-independent witnessing dimensions greater than two. Furthermore, the algorithm with some modifications may find applications in various black-box quantum information tasks with large number of inputs and outputs.
AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant KGK_GKG​, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of KGK_GKG​ is not known, we recently tightened the best known bounds to 6π11  ≤  KG  ≤  π2log⁡(1+2)−10−4. \frac{6π}{11} \;\le\; K_G \;\le\; \fracπ{2\log(1+\sqrt2)} - 10^{-4}. 116π​≤KG​≤2log(1+2​)π​−10−4. Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.