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 $A=(a_{ij})\in {\mathbb{C}}^{m\times n}$,

$$\underset{\begin{array}{c}{u}_1,\dots ,u_m\in S^{\infty }\\ v_1,\dots ,v_n\in S^{\infty }\end{array}}{\max }\left|\sum _{i=1}^m\sum _{j=1}^n{a}_{ij}⟨u_i,v_j⟩\right|\le C_{10b}\underset{\begin{array}{c}|s_1|=\cdots =|s_m|=1\\ |t_1|=\cdots =|t_n|=1\end{array}}{\max }\left|\sum _{i=1}^m\sum _{j=1}^n{a}_{ij}{s}_i{t}_j\right|.$$

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

Known lower bounds

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 $1.40458\dots$ 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• [D1984] Davie, A. M. Private communication / unpublished note (1984).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 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.