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Grothendieck's Constant

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The real Grothendieck constant of order n, denoted k_R(n), is defined by allowing arbitrary finite matrix sizes while restricting the vectors to R^n. Let A be an m×m real matrix, where m is a positive integer, such that

|sum_(i=1)^msum_(j=1)^ma_(ij)s_it_j|<=1
(1)

for all real numbers s_1, ..., s_m and t_1, ..., t_m satisfying |s_i|,|t_j|<=1. The constant k_R(n) is the smallest number for which

|sum_(i=1)^msum_(j=1)^ma_(ij)x_i·y_j|<=k_R(n)
(2)

holds for every such matrix and all vectors x_i,y_j in R^n with ||x_i||,||y_j||<=1. Rectangular matrices give the same constant by adding zero rows or columns to obtain a square matrix (Designolle et al. 2026). Grothendieck (1953) proved that these constants have a finite universal upper bound independent of n.

For example, bounds for small n are

k_R(2)=sqrt(2) 1.43670<=k_R(3)<=1.4546 1.48579<=k_R(4)<=1/2pi approx 1.5708
(3)

(Krivine 1977, 1979; König 1992; Finch 2003, p. 236; Designolle et al. 2023, 2026).

Now consider the limit

k_R=lim_(n->infty)k_R(n),
(4)

which is sometimes also denoted K_G. Classical bounds include

1.67696...<=k_R<=1.7822139781...,
(5)

and Krivine (1977, 1979) postulated that

k_R=pi/(2ln(1+sqrt(2)))=1.7822139...
(6)

(OEIS A088367; Finch 2003, p. 236). The conjecture was refuted in 2011 by Yury Makarychev, Mark Braverman, Konstantin Makarychev, and Assaf Naor, who showed that k_R is strictly less than Krivine's bound (Makarychev 2011, Braverman et al. 2013).

Heilman et al. (2026a) reported the improved real bounds

1.773<=k_R<=1.7799.
(7)

Their lower bound is certified using a finite optimization certificate and their upper bound using interval arithmetic. A numerical estimate near 1.779 is heuristic and is not a proved value. Heilman et al. (2026a) report AI assistance with the search and certification. The certificates have not been independently rerun here.

Peng (2026) improved the real universal upper bound to

k_R<(44496791)/(25000000)=1.77987164.
(8)

The proof applies to all finite matrix sizes and vector dimensions and combines interval arithmetic with an exact rational certificate. Peng (2026) reports substantive assistance from OpenAI Codex with the argument and certification.

Similarly, if the numbers s_i and t_j and matrix A are complex and the vectors are in C^n, then the corresponding constants k_C(n) are defined using the complex inner product. These are known to satisfy

k_C(2) in [1.1526,1.2157]
(9)
k_C(3) in [1.2108,1.2744]
(10)
k_C(4) in [1.2413,1.3048]
(11)

(Krivine 1977, 1979; König 1990, 1992; Finch 2003, p. 236).

The limit

k_C=lim_(n->infty)k_C(n)
(12)

satisfies

1.35584631827168<k_C<=1.40491
(13)

(Krivine 1977, 1979; Haagerup 1987; Finch 2003, p. 236; Guo et al. 2026), where Haagerup's upper bound (OEIS A088374) is given by 8/[pi(x_0+1)] with

psi(x)=xint_0^(pi/2)(cos^2theta)/(sqrt(1-x^2sin^2theta))dtheta
(14)
=1/x[E(x)-(1-x^2)K(x)],
(15)

E(k) a complete elliptic integral of the second kind, K(k) a complete elliptic integral of the first kind, and x_0=0.812557... (OEIS A088373) the root of

psi(x)=1/8pi(x+1).
(16)

Haagerup (1987) doubted that this upper bound is the exact value of k_C and proposed the smaller candidate

(int_0^(pi/2)(cos^2theta)/(sqrt(1+sin^2theta))dtheta)^(-1)=1/(2K(i)-E(i))
(17)
=1.4045759...
(18)

(OEIS A088375; Finch 2003, pp. 236-237). Guo et al. (2026) obtained the strict lower bound using an analytic reduction followed by numerical inequalities certified using interval arithmetic. The authors credit the Odin Automatic AI Research Agent with discovering the lower bound and proof. Independent review of the analytic argument had not been reported as of Sep. 12, 2026.

Heilman et al. (2026b) reported the sharper complex bounds

1.4<k_C<1.404898554746.
(19)

Their proof reduces both bounds to inequalities certified by numerical computation. The supporting certificates have not been independently rerun here. Haagerup's proposed exact value remains a conjecture.

See also

Bell Inequality, CHSH Inequality, Hilbert Space

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References

Braverman, M.; Makarychev, K.; Makarychev, Y.; and Naor, A. "The Grothendieck Constant Is Strictly Smaller than Krivine's Bound." Forum Math. Pi 1, e4, 2013. https://doi.org/10.1017/fmp.2013.4.Designolle, S.; Iommazzo, G.; Besançon, M.; Knebel, S.; Gelß, P.; and Pokutta, S. "Improved Local Models and New Bell Inequalities via Frank-Wolfe Algorithms." Phys. Rev. Res. 5, 043059, 2023. https://doi.org/10.1103/PhysRevResearch.5.043059.Designolle, S.; Vértesi, T.; and Pokutta, S. "Better Bounds on Finite-Order Grothendieck Constants with Applications to Bell Nonlocality." Phys. Rev. A 113, 022401, 2026. https://doi.org/10.1103/m5mn-c5dx.Finch, S. R. "Grothendieck's Constants." §3.11 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 235-237, 2003.Fishburn, P. C. and Reeds, J. A. "Bell Inequalities, Grothendieck's Constant, and Root Two." SIAM J. Discr. Math. 7, 48-56, 1994.Grothendieck, A. "Résumé de la théorie métrique des produits tensoriels topologiques." Bol. Soc. Mat. Sao Paulo 8, 1-79, 1953.Guo, S.; Fang, E. X.; and Lu, J. "An Improved Lower Bound for the Complex Grothendieck Constant." 7 Sep 2026. https://arxiv.org/abs/2609.07000.Haagerup, U. "A New Upper Bound for the Complex Grothendieck Constant." Israel J. Math. 60, 199-224, 1987. https://doi.org/10.1007/BF02790792.Heilman, S.; Jones, C.; and Malavolta, G. "Primal/Dual Method for the Grothendieck Constant." 7 Oct 2026a. https://arxiv.org/abs/2610.10477.Heilman, S.; Jones, C.; and Malavolta, G. "Sharper Bounds for the Complex Grothendieck Constant." 21 Sep 2026b. https://arxiv.org/abs/2609.25462.Jameson, G. L. O. Summing and Nuclear Norms in Banach Space Theory. Cambridge, England: Cambridge University Press, 1987.König, H. "On the Complex Grothendieck Constant in the n-Dimensional Case." In Geometry of Banach Spaces: Proceedings of the Conference Held in Linz, 1989 (Ed. P. F. X. Müller and W. Schachermayer). Cambridge, England: Cambridge University Press, pp. 181-198, 1990.König, H. "Some Remarks on the Grothendieck Inequality." General Inequalities 6, Proc. 1990 Oberwolfach Conference (Ed. W. Walter). Basel, Switzerland: Birkhäuser, pp. 201-206, 1992.Krivine, J.-L. "Sur la constante de Grothendieck." C. R. Acad. Sci. Paris 284, 445-446, 1977.Krivine, J.-L. "Constantes de Grothendieck et fonctions de type positif sur les sphères." Adv. Math. 31, 16-30, 1979.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 42, 1983.Lindenstrauss, J. and Pełczyński, A. "Absolutely Summing Operators in L_p Spaces and Their Applications." Studia Math. 29, 275-326, 1968.Makarychev, Y. "The Grothendieck Constant Is Strictly Smaller than Krivine." Seminar. Cambridge, MA: MIT Computer Science and Artificial Intelligence Laboratory. Nov. 8, 2011.Sloane, N. J. A. Sequences A088367, A088373, A088374, and A088375 in "The On-Line Encyclopedia of Integer Sequences."Peng, B. "Certified Fourier Rounding for the Real Grothendieck Inequality." 17 Oct 2026. https://arxiv.org/abs/2610.24417.

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Grothendieck's Constant

Cite this as:

Weisstein, Eric W. "Grothendieck's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrothendiecksConstant.html

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