The exact real Grothendieck constant
The real Grothendieck constant $K_G$ is the smallest real number such that, for all positive integers $m, n, d$, every real matrix $A = (a_{ij}) \in \mathbb{R}^{m \times n}$ and all unit vectors $u_i, v_j \in \mathbb{R}^d$, $$\Bigl| \sum_{i=1}^m \sum_{j=1}^n a_{ij} \langle u_i, v_j \rangle \Bigr| \le K_G \max_{\epsilon_i, \delta_j \in \{-1, 1\}} \Bigl| \sum_{i=1}^m \sum_{j=1}^n a_{ij} \epsilon_i \delta_j \Bigr|.$$ Grothendieck proved $\pi/2 \le K_G \le \sinh(\pi/2)$. Krivine proved $K_G \le \pi/(2\log(1 + \sqrt{2})) = 1.782\ldots$ and conjectured that this is the exact value; Braverman, Makarychev, Makarychev and Naor proved that the inequality is strict. Davie and Reeds independently proved $K_G \ge 1.6769\ldots$, and Heilman and, separately, Jones and Malavolta showed that this lower bound is not optimal. Saha, Li, Xue, Chaudhuri, Klivans, Kothari and Meka proved $$\frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1 + \sqrt{2})} - 10^{-4}$$ (2026).
Problem. Determine the exact value of $K_G$.
Source: Grothendieck (1953), who proved π/2 ≤ K_G ≤ sinh(π/2); Krivine (1977) conjectured K_G = π/(2 log(1 + √2)), which Braverman, Makarychev, Makarychev and Naor (2013) disproved. See Pisier (2012), §4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Grothendieck inequality, semidefinite relaxations, sharp constants
References
@article{grothendieck1953,
author = {Grothendieck, Alexandre},
title = {R{\'e}sum{\'e} de la th{\'e}orie m{\'e}trique des produits tensoriels topologiques},
journal = {Bol. Soc. Mat. S{\~a}o Paulo},
volume = {8},
year = {1953},
pages = {1--79},
}
@article{krivine1977,
author = {Krivine, Jean-Louis},
title = {Sur la constante de {G}rothendieck},
journal = {C. R. Acad. Sci. Paris S{\'e}r. A-B},
volume = {284},
number = {8},
year = {1977},
pages = {A445--A446},
}
@article{braverman2013,
author = {Braverman, Mark and Makarychev, Konstantin and Makarychev, Yury and Naor, Assaf},
title = {The {G}rothendieck constant is strictly smaller than {K}rivine's bound},
journal = {Forum Math. Pi},
volume = {1},
year = {2013},
pages = {e4},
doi = {10.1017/fmp.2013.4},
eprint = {1103.6161},
archivePrefix = {arXiv},
}
@article{pisier2012,
author = {Pisier, Gilles},
title = {Grothendieck's theorem, past and present},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {49},
number = {2},
year = {2012},
pages = {237--323},
doi = {10.1090/S0273-0979-2011-01348-9},
eprint = {1101.4195},
archivePrefix = {arXiv},
}
@misc{heilman2026,
author = {Heilman, Steven},
title = {A lower bound for {G}rothendieck's constant},
year = {2026},
eprint = {2603.22616},
archivePrefix = {arXiv},
}
@misc{jones2026,
author = {Jones, Chris and Malavolta, Giulio},
title = {The {G}rothendieck constant is strictly larger than {D}avie--{R}eeds' bound},
year = {2026},
eprint = {2603.30039},
archivePrefix = {arXiv},
}
@misc{saha2026,
author = {Saha, Rahul and Li, Alan and Xue, Anton and Chaudhuri, Swarat and Klivans, Adam and Kothari, Pravesh K. and Meka, Raghu},
title = {New lower and upper bounds for the {G}rothendieck constant},
year = {2026},
eprint = {2608.11158},
archivePrefix = {arXiv},
}
@misc{aim-real-grothendieck-constant,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The exact real Grothendieck constant}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/027-real-grothendieck-constant.md},
note = {Entry 027 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00376, "The exact real Grothendieck constant", clopen.solutions.