The complex Grothendieck constant
Description of constant
The complex Grothendieck constant (often denoted $K_G^{\mathbb{C}}$) is the smallest number $C_{10b}$ such that, for every $m,n\ge 1$ and every complex matrix $A=(a_{ij})\in\mathbb{C}^{m\times n}$,
\[\max_{\substack{u_1,\dots,u_m\in S^{\infty}\\ v_1,\dots,v_n\in S^{\infty}}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}\langle u_i, v_j\rangle\right| \ \le\ C_{10b}\ \max_{\substack{|s_1|=\cdots=|s_m|=1\\ |t_1|=\cdots=|t_n|=1}} \left|\sum_{i=1}^m\sum_{j=1}^n a_{ij}s_it_j\right|.\]Here $S^{\infty}$ denotes the unit sphere in a (complex) Hilbert space, $\langle\cdot,\cdot\rangle$ is the Hermitian inner product, and $t_{1}, \ldots, t_{n}, s_{1}, \ldots, s_{m}$ are complex numbers.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1.607$ | [Kai1973] | Bound via the method of Rietz (as cited by Haagerup). |
| $e^{1-\gamma}\approx 1.52621$ | [P1978] | Here $\gamma$ 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, $C_{10b}$ 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 $1.40458\ldots$ 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- [D1984] Davie, A. M. Private communication / unpublished note (1984).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.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.