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The real Grothendieck constant

Description of constant

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

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

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

$$ \max_{\substack{u_1,\dots,u_{m}, v_{1},\dots,v_{n} \in S^{\infty}}} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \langle u_{i}, v_{j}\rangle \ \le\\ C \max_{\varepsilon_{1},\dots,\varepsilon_{m}, \delta_{1},\dots,\delta_{n} = \pm 1} \ \sum_{i=1}^m \sum_{j=1}^n a_{ij} \varepsilon_{i} \delta_{j}. $$

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
$\sinh(\pi/2) \approx 2.30130$ [G1953] Grothendieck’s original upper bound
$2.261$ [R1974] Improvement of the original upper bound
$\dfrac{\pi}{2\ln(1+\sqrt{2})} \approx 1.782214$ [K1979] Krivine’s bound; best known explicit numerical upper bound
$< \dfrac{\pi}{2\ln(1+\sqrt{2})}$ [BMMN2011] Strict improvement over Krivine’s bound (no widely cited explicit numerical gap)

Known lower bounds

Bound Reference Comments
$1$ Trivial Follows from the definitions
$\dfrac{\pi}{2} \approx 1.57080$ [G1953] Grothendieck’s original lower bound
$1.67696\ldots$ [Dav1984], [Ree1991] (due to Davie and independently Reeds)
$1.67696\ldots + 10^{-26}$ [Hei26] Strictly improves over Davie--Reeds
$1.67696\ldots + 10^{-12}$ [JM26] Best known lower bound, concurrent with [Hei26]

Additional comments and links

  • Krivine conjectured that $C_{10} = \frac{\pi}{2\ln(1+\sqrt{2})}$, but this was disproved in [BMMN2011] by showing the inequality is strict. It seems that [BMMN2011] proves that $K_{G}< C_{10} - 10^{-500}$.
  • A standard reference survey is [Pis2012].
  • Wikipedia page on the Grothendieck inequality
  • The strategy of [Hei26] could possibly be used to improve the complex Grothendieck constant lower bound.

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 Unpublished note (1984).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
  • [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.
  • [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.