The complex Grothendieck constant

The complex Grothendieck constant (often denoted K_G^ℂ) is the smallest number C_10b such that, for every m,n≥ 1 and every complex matrix A=(a_ij)∈ℂ^m× n, max_substacku_1,…,u_m∈ S^∞\ v_1,…,v_n∈ S^∞ |Σ_i=1^mΣ_j=1^n a_ij⟨ u_i, v_j⟩| ≤ C_10b\ max_substack|s_1|=⋯=|s_m|=1\ |t_1|=⋯=|t_n|=1…

Current state (tracked automatically)

Best known bounds for constant 10b: 1.338 ≤ C ≤ 1.40491. Upper bound ([H1987], Best known general upper bound (Haagerup)); lower bound ([D1984], Best known general lower bound (Davie).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 10b)	1.338 [D1984] — Best known general lower bound (Davie;	Tao's optimization constants
Best upper bound with an explicit value (constant 10b)	1.40491 [H1987] — Best known general upper bound (Haagerup).	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

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Cite

@misc{cairn-constant-10b-the-complex-grothendieck-constant,
  title        = {The complex Grothendieck constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-10b-the-complex-grothendieck-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-13}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

The complex Grothendieck constant (often denoted $K_{G}^{C}$ ) is the smallest number $C_{10 b}$ such that, for every $m, n \geq 1$ and every complex matrix $A = (a_{ij}) \in C^{m \times n}$ ,

$u_{1}, \dots, u_{m} \in S^{\infty} v_{1}, \dots, v_{n} \in S^{\infty} max i = 1 \sum m j = 1 \sum n a_{ij} ⟨ u_{i}, v_{j} ⟩ \leq C_{10 b} ∣ s_{1} ∣ = \dots = ∣ s_{m} ∣ = 1 ∣ t_{1} ∣ = \dots = ∣ t_{n} ∣ = 1 max i = 1 \sum m j = 1 \sum n a_{ij} s_{i} t_{j} .$

Here $S^{\infty}$ denotes the unit sphere in a (complex) Hilbert space, $⟨ \cdot, \cdot ⟩$ is the Hermitian inner product, and $t_{1}, \dots, t_{n}, s_{1}, \dots, s_{m}$ are complex numbers.

Known upper bounds

Bound	Reference	Comments
$1.607$	[Kai1973]	Bound via the method of Rietz (as cited by Haagerup).
$e^{1 - γ} \approx 1.52621$	[P1978]	Here $γ$ is the Euler--Mascheroni constant.
$1.40491$	[H1987]	Best known general upper bound (Haagerup).

Known lower bounds

Bound	Reference	Comments
$1$	Trivial
$1.338$	[D1984]	Best known general lower bound (Davie; cited by Haagerup).

Additional comments and links

In optimization terms, $C_{10 b}$ is the worst-case ratio between the natural semidefinite relaxation (vectors in a Hilbert space) and the original “phase” optimization (scalars of modulus $1$ ) for bilinear forms with complex coefficients.
Haagerup suggested a plausible (conjectural) slightly smaller value $1.40458 \dots$ in [H1987] (unproved).
Wikipedia page on Grothendieck inequality

References

[D1984] Davie, A. M. Private communication / unpublished note (1984). (Cited in [H1987].)
[G1953] Grothendieck, A. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8 (1956), 1–79. (Originally written 1953.)
[H1987] Haagerup, U. A new upper bound for the complex Grothendieck constant. Israel J. Math. 60 (1987), no. 2, 199–224.
[Kai1973] Kaijser, S. A note on the Grothendieck constant with an application to harmonic analysis. UUDM Report No. 1973:10, Uppsala University (mimeographed).
[P1978] Pisier, G. Grothendieck's theorem for non-commutative C-algebras with an appendix on Grothendieck's constant.* J. Funct. Anal. 29 (1978), 379–415.

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 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.