From 4124cd5a9a6be67bc82298395051ecec42e91dad Mon Sep 17 00:00:00 2001 From: sheilman77 Date: Tue, 2 Jun 2026 20:56:28 -0700 Subject: [PATCH 1/3] Update 10a.md --- constants/10a.md | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/constants/10a.md b/constants/10a.md index cd8e29b..0b99298 100644 --- a/constants/10a.md +++ b/constants/10a.md @@ -28,6 +28,8 @@ $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$). | $2.261$ | [R1974] | Improvement of the original upper bound | | $\dfrac{\pi}{2\ln(1+\sqrt{2})} \approx 1.782214$ | [K1979] | Krivine’s bound; best known **explicit** numerical upper bound | | $< \dfrac{\pi}{2\ln(1+\sqrt{2})}$ | [BMMN2011] | Strict improvement over Krivine’s bound (no widely cited explicit numerical gap) | +| $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-5}$ | [Hei26b] | $10^{-5}$ improvement over Krivine's bound | +| $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039*10^{-5}$ | [LSXKKMC26] | $6.039*10^{-5}$ improvement over Krivine's bound | ## Known lower bounds @@ -46,6 +48,7 @@ $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$). - [Wikipedia page on the Grothendieck inequality](https://en.wikipedia.org/wiki/Grothendieck_inequality) - [JM26] proves that the Davie–Reeds lower bound is not optimal, using a perturbative analysis of the Davie–Reeds operator to establish $K_{G}^{\mathbb R} \ge K_{DR} + 10^{-12}$; the paper also notes concurrent work [Hei26] proving the weaker bound $K_{G}^{\mathbb R} \ge K_{DR} + 10^{-26}$. - 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. [LSXKKMC26] upgrades this to a degree 9 thresholding mixed with hyerplane thresholding. ## References @@ -53,8 +56,10 @@ $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$). - [Dav1984] Davie, A. M. *Lower bound for $K_{G}$.* Unpublished note (1984).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](https://arxiv.org/abs/2603.22616) +- [Hei26b] Heilman, Steven. *An Upper Bound on Grothendieck's constant.* (2026) [arXiv:2606.00247](https://arxiv.org/abs/2606.00247) - [JM26] Jones, Chris; Malavolta, Giulio. *The Grothendieck constant is strictly larger than Davie-Reeds' bound.* (2026) [arXiv:2603.30039](https://arxiv.org/abs/2603.30039) - [K1979] Krivine, Jean-Louis. *Constantes de Grothendieck et fonctions de type positif sur les sphères.* Advances in Mathematics **31** (1979), 16–30. +- [LSXKKMC26], Alan Li, Rahul Saha, Anton Xue, Adam Klivans, Pravesh K Kothari, Raghu Meka, Swarat Chaudhury. *The Grothendieck Constant is Less Than $\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-5}$* [arXiv:2606.03991](https://arxiv.org/abs/2606.03991) - [Pis2012] Pisier, Gilles. *Grothendieck’s theorem, past and present.* Bull. Amer. Math. Soc. (N.S.) **49** (2012), 237–323. [arXiv:1101.4195](https://arxiv.org/abs/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. From c1b6cbc1fcdd5d39572d45c3f0d433484960faad Mon Sep 17 00:00:00 2001 From: sheilman77 Date: Tue, 2 Jun 2026 20:57:06 -0700 Subject: [PATCH 2/3] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 7ad6708..95ed173 100644 --- a/README.md +++ b/README.md @@ -29,7 +29,7 @@ Bounds for which the level of available verification is currently at minimal lev | [7b](https://teorth.github.io/optimizationproblems/constants/7b.html) | Irrationality measure of $\Gamma(1/4)$ | 2 | $10^{143}$ | | [8](https://teorth.github.io/optimizationproblems/constants/8a.html) | Classical zero-free region constant | 0.755106 | 5.558691 | | [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.2578 | 3.3177 | -| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.782214 | +| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.78221397819137 | | [10b](https://teorth.github.io/optimizationproblems/constants/10b.html) | The complex Grothendieck constant | 1.338 | 1.40491 | | [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.414214 | 3.674235 (3.65*) | | [11a](https://teorth.github.io/optimizationproblems/constants/11a.html) | $L^1$ Poincaré constant on the Hamming cube | $\sqrt{\pi/2} \approx 1.2533$ | $\pi/2 - 0.00013 \approx 1.5707$ | From 2c0be48a7dedc059b32644d5d3079a62c8d156f3 Mon Sep 17 00:00:00 2001 From: sheilman77 Date: Tue, 2 Jun 2026 20:57:43 -0700 Subject: [PATCH 3/3] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 95ed173..4094b6b 100644 --- a/README.md +++ b/README.md @@ -29,7 +29,7 @@ Bounds for which the level of available verification is currently at minimal lev | [7b](https://teorth.github.io/optimizationproblems/constants/7b.html) | Irrationality measure of $\Gamma(1/4)$ | 2 | $10^{143}$ | | [8](https://teorth.github.io/optimizationproblems/constants/8a.html) | Classical zero-free region constant | 0.755106 | 5.558691 | | [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.2578 | 3.3177 | -| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.78221397819137 | +| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.78215358819137 | | [10b](https://teorth.github.io/optimizationproblems/constants/10b.html) | The complex Grothendieck constant | 1.338 | 1.40491 | | [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.414214 | 3.674235 (3.65*) | | [11a](https://teorth.github.io/optimizationproblems/constants/11a.html) | $L^1$ Poincaré constant on the Hamming cube | $\sqrt{\pi/2} \approx 1.2533$ | $\pi/2 - 0.00013 \approx 1.5707$ |