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OP-12009OpenAnalysis

The complex Grothendieck constant (C10bC_{10b})

The complex Grothendieck constant: determine C10bC_{10b} or improve its bounds; the repository records 1.338 (lower) and 1.40491 (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

The complex Grothendieck constant (often denoted KGCK_G^{\mathbb{C}}) is the smallest number C10bC_{10b} such that, for every m,n≥1m,n\ge 1 and every complex matrix A=(aij)∈Cm×nA=(a_{ij})\in\mathbb{C}^{m\times n},

max⁡u1,…,um∈S∞v1,…,vn∈S∞∣∑i=1m∑j=1naij⟨ui,vj⟩∣ ≤ C10b max⁡∣s1∣=⋯=∣sm∣=1∣t1∣=⋯=∣tn∣=1∣∑i=1m∑j=1naijsitj∣.\max_{\substack{u_1,\dots,u_m\in S^{\infty}\\ v_1,\dots,v_n\in S^{\infty}}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}\langle u_i, v_j\rangle\right| \ \le\ C_{10b}\ \max_{\substack{|s_1|=\cdots=|s_m|=1\\ |t_1|=\cdots=|t_n|=1}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}s_it_j\right|.

Here S∞S^{\infty} denotes the unit sphere in a (complex) Hilbert space, ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle is the Hermitian inner product, and t1,…,tn,s1,…,smt_{1}, \ldots, t_{n}, s_{1}, \ldots, s_{m} are complex numbers.

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Background

  • In optimization terms, C10bC_{10b} is the worst-case ratio between the natural semidefinite relaxation (vectors in a Hilbert space) and the original “phase” optimization (scalars of modulus 11) for bilinear forms with complex coefficients.
  • Haagerup suggested a plausible (conjectural) slightly smaller value 1.40458…1.40458\ldots in [H1987] (unproved).
  • Wikipedia page on Grothendieck inequality

References (from the repository page)

  • [D1984] Davie, A. M. Private communication / unpublished note (1984).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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.338; best known upper bound 1.40491. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1.6071.607 [Kai1973] Bound via the method of Rietz (as cited by Haagerup).
e1−γ≈1.52621e^{1-\gamma}\approx 1.52621 [P1978] Here γ\gamma is the Euler--Mascheroni constant.
1.404911.40491 [H1987] Best known general upper bound (Haagerup).

Known lower bounds

Bound Reference Comments
11 Trivial
1.3381.338 [D1984] Best known general lower bound (Davie; cited by Haagerup).
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — The complex Grothendieck constant ($C_{10b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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