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OP-12008OpenAnalysis

The real Grothendieck constant (C10aC_{10a})

The real Grothendieck constant: determine C10aC_{10a} or improve its bounds; the repository records 6π11≈1.71360\frac{6\pi}{11}\approx 1.71360 (lower) and π2log⁡(1+2)−10−4≈1.78211\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4} \approx 1.78211 (upper).

MSC 46B28 · Spaces of operators; tensor products; approximation properties

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

C10C_{10} is the real Grothendieck constant KGRK_{G}^{\mathbb R}.

It is the smallest constant CC such that for every m,n≥1m,n \ge 1 and every real matrix

A=(aij)∈Rm×nA=(a_{ij}) \in \mathbb{R}^{m\times n} one has

max⁡u1,…,um,v1,…,vn∈S∞ ∑i=1m∑j=1naij⟨ui,vj⟩ ≤ Cmax⁡ε1,…,εm,δ1,…,δn=±1 ∑i=1m∑j=1naijεiδj.\max_{\substack{u_1,\dots,u_{m}, v_{1},\dots,v_{n} \in S^{\infty}}} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \langle u_{i}, v_{j}\rangle \ \le\ C \max_{\varepsilon_{1},\dots,\varepsilon_{m}, \delta_{1},\dots,\delta_{n} = \pm 1} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \varepsilon_{i} \delta_{j}.

Here S∞S^{\infty} denotes the unit sphere of a real Hilbert space (equivalently, one may take ui,vj∈Sd−1u_{i},v_{j} \in S^{d-1} for some sufficiently large finite dimension dd).

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Background

  • Krivine conjectured that C10=π2ln⁡(1+2)C_{10} = \frac{\pi}{2\ln(1+\sqrt{2})}, but this was disproved in [BMMN2011] by showing the inequality is strict.
  • A standard reference survey is [Pis2012].
  • Wikipedia page on the Grothendieck inequality
  • [JM26] proves that the Davie–Reeds lower bound is not optimal, using a perturbative analysis of the Davie–Reeds operator to establish KGR≥KDR+10−12K_{G}^{\mathbb R} \ge K_{DR} + 10^{-12}; the paper also notes concurrent work [Hei26] proving the weaker bound KGR≥KDR+10−26K_{G}^{\mathbb R} \ge K_{DR} + 10^{-26}.
  • The perturbative strategy of [Hei26] and [JM26] could possibly be used to improve the complex Grothendieck constant lower bound.
  • [Hei26b] uses thresholding by degree 3 Hermite polynomials. [LSXCKKM26] upgrades this to a degree 9 thresholding mixed with hyperplane thresholding.
  • [SLXCKKM26] (by the same 7-author team as [LSXCKKM26], with author order permuted so Saha appears first) settles the tenths digit of KGRK_G^{\mathbb R}: 1.71360≤KGR≤1.782121.71360 \le K_G^{\mathbb R} \le 1.78212, so the first decimal digit is 77. Both bounds are asymptotic — the lower bound is the first not obtained via an explicit low-dimensional gap instance, and the upper bound is the first obtained via an asymptotic (rather than low-dimensional) Krivine-type rounding scheme. The authors state that the bounds "were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered".

References (from the repository page)

  • [BMMN2011] Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf. The Grothendieck constant is strictly smaller than Krivine's bound. Forum of Mathematics, Pi, Volume 1, 2013, e4. arXiv:1103.6161
  • [Dav1984] Davie, A. M. Lower bound for KGK_{G}. Unpublished note (1984).
  • [G1953] Grothendieck, Alexandre. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 (1953), 1–79.
  • [Hei26] Heilman, Steven. A lower bound for Grothendieck's constant. (2026) arXiv:2603.22616
  • [Hei26b] Heilman, Steven. An Upper Bound on Grothendieck's constant. (2026) arXiv:2606.00247
  • [JM26] Jones, Chris; Malavolta, Giulio. The Grothendieck constant is strictly larger than Davie-Reeds' bound. (2026) arXiv:2603.30039
  • [K1979] Krivine, Jean-Louis. Constantes de Grothendieck et fonctions de type positif sur les sphères. Advances in Mathematics 31 (1979), 16–30.
  • [LSXCKKM26] Li, Alan; Saha, Rahul; Xue, Anton; Chaudhuri, Swarat; Klivans, Adam; Kothari, Pravesh K; Meka, Raghu. The Grothendieck Constant is Less Than π2log⁡(1+2)−10−5\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-5} arXiv:2606.03991
  • [SLXCKKM26] Saha, Rahul; Li, Alan; Xue, Anton; Chaudhuri, Swarat; Klivans, Adam; Kothari, Pravesh K; Meka, Raghu. New Lower and Upper Bounds for the Grothendieck Constant. arXiv:2608.11158 (2026).
  • [Pis2012] Pisier, Gilles. Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.) 49 (2012), 237–323. arXiv:1101.4195
  • [Ree1991] Reeds, James A. A new lower bound on the real Grothendieck constant. Unpublished manuscript (1991).
  • [R1974] Rietz, Ronald E. A proof of the Grothendieck inequality. Israel J. Math. 19 (1974), 271–276.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 6π11≈1.71360\frac{6\pi}{11}\approx 1.71360; best known upper bound π2log⁡(1+2)−10−4≈1.78211\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4} \approx 1.78211. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
sinh⁡(π/2)≈2.30130\sinh(\pi/2) \approx 2.30130 [G1953] Grothendieck’s original upper bound
2.2612.261 [R1974] Improvement of the original upper bound
π2ln⁡(1+2)≈1.782214\dfrac{\pi}{2\ln(1+\sqrt{2})} \approx 1.782214 [K1979] Krivine’s bound; best known explicit numerical upper bound
<π2ln⁡(1+2)< \dfrac{\pi}{2\ln(1+\sqrt{2})} [BMMN2011] Strict improvement over Krivine’s bound
<π2ln⁡(1+2)−10−5< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-5} [Hei26b] 10−510^{-5} improvement over Krivine's bound
<π2ln⁡(1+2)−6.039×10−5< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039\times 10^{-5} [LSXCKKM26] 6.039×10−56.039\times 10^{-5} improvement over Krivine's bound
<π2ln⁡(1+2)−10−4< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-4} [SLXCKKM26] Same team; via the first asymptotic Krivine-type rounding scheme (previous works only considered low-dimensional schemes).

Known lower bounds

Bound Reference Comments
11 Trivial Follows from the definitions
π2≈1.57080\dfrac{\pi}{2} \approx 1.57080 [G1953] Grothendieck’s original lower bound
KDR≈1.67696…K_{DR} \approx 1.67696\ldots [Dav1984], [Ree1991] Davie–Reeds lower bound
KDR+10−26K_{DR} + 10^{-26} [Hei26] Concurrent 2026 improvement over the Davie–Reeds bound
KDR+10−12K_{DR} + 10^{-12} [JM26] Strict improvement over the Davie–Reeds bound
6π11≈1.71360\dfrac{6\pi}{11}\approx 1.71360 [SLXCKKM26] Major jump over the Davie–Reeds anchor ≈1.67696\approx 1.67696; proved by establishing asymptotic limitations of Krivine schemes rather than by constructing explicit gap instances (as all prior lower bounds did). Together with the upper bound above, this determines the tenths digit of KGRK_G^{\mathbb R} to be 77.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — The real Grothendieck constant ($C_{10a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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