The real Grothendieck constant

Description of constant

$C_{10}$ is the real Grothendieck constant $K_{G}^{R}$ .

It is the smallest constant $C$ such that for every $m, n \geq 1$ and every real matrix

$A = (a_{ij}) \in R^{m \times n}$ one has

$u_{1}, \dots, u_{m}, v_{1}, \dots, v_{n} \in S^{\infty} max i = 1 \sum m j = 1 \sum n a_{ij} ⟨ u_{i}, v_{j} ⟩ \leq C ε_{1}, \dots, ε_{m}, δ_{1}, \dots, δ_{n} = \pm 1 max i = 1 \sum m j = 1 \sum n a_{ij} ε_{i} δ_{j} .$

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

Known upper bounds

Bound	Reference	Comments
$sinh (π /2) \approx 2.30130$	[G1953]	Grothendieck’s original upper bound
$2.261$	[R1974]	Improvement of the original upper bound
$\frac{π}{2 ln ( 1 + 2 )} \approx 1.782214$	[K1979]	Krivine’s bound; best known explicit numerical upper bound
$< \frac{π}{2 ln ( 1 + 2 )}$	[BMMN2011]	Strict improvement over Krivine’s bound
$< \frac{π}{2 ln ( 1 + 2 )} - 1 0^{- 5}$	[Hei26b]	$1 0^{- 5}$ improvement over Krivine's bound
$< \frac{π}{2 ln ( 1 + 2 )} - 6.039 \times 1 0^{- 5}$	[LSXCKKM26]	$6.039 \times 1 0^{- 5}$ improvement over Krivine's bound
$< \frac{π}{2 ln ( 1 + 2 )} - 1 0^{- 4}$	[SLXCKKM26]	Same team; via the first asymptotic Krivine-type rounding scheme (previous works only considered low-dimensional schemes).

Known lower bounds

Bound	Reference	Comments
$1$	Trivial	Follows from the definitions
$\frac{π}{2} \approx 1.57080$	[G1953]	Grothendieck’s original lower bound
$K_{D R} \approx 1.67696 \dots$	[Dav1984], [Ree1991]	Davie–Reeds lower bound
$K_{D R} + 1 0^{- 26}$	[Hei26]	Concurrent 2026 improvement over the Davie–Reeds bound
$K_{D R} + 1 0^{- 12}$	[JM26]	Strict improvement over the Davie–Reeds bound
$\frac{6 π}{11} \approx 1.71360$	[SLXCKKM26]	Major jump over the Davie–Reeds anchor $\approx 1.67696$ ; 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 $K_{G}^{R}$ to be $7$ .

Additional comments and links

Krivine conjectured that $C_{10} = \frac{π}{2 l n ( 1 + 2 )}$ , 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 $K_{G}^{R} \geq K_{D R} + 1 0^{- 12}$ ; the paper also notes concurrent work [Hei26] proving the weaker bound $K_{G}^{R} \geq K_{D R} + 1 0^{- 26}$ .
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 $K_{G}^{R}$ : $1.71360 \leq K_{G}^{R} \leq 1.78212$ , so the first decimal digit is $7$ . 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

[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 $K_{G}$ . 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
[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 $\frac{π}{2 l o g ( 1 + 2 )} - 1 0^{- 5}$ 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.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem