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Grothendieck's Constant

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Grothendieck's constant is the least universal constant in the matrix inequality below. Let A be an n×n real square matrix with n>=2 such that

|sum_(i=1)^nsum_(j=1)^na_(ij)s_it_j|<=1
(1)

for all real numbers s_1, s_2, ..., s_n and t_1, t_2, ..., t_n such that |s_i|,|t_j|<=1. Then Grothendieck (1953) showed that there exists a constant k_R(n) satisfying

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

for all vectors x_1,x_2,...,x_m and y_1,y_2,...,y_n in a Hilbert space with norms ||x_i||<=1 and ||y_j||<=1.

The Grothendieck constant is the smallest possible value of k_R(n). For example, the best values known for small n are

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

(Krivine 1977, 1979; König 1992; Finch 2003, p. 236; Designolle et al. 2023, 2024).

Now consider the limit

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

which is related to Khinchin's constant and sometimes also denoted K_G. Krivine (1977) showed that

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

and postulated that

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

(OEIS A088367). 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).

Similarly, if the numbers s_i and t_j and matrix A are taken as complex, then a similar set of constants k_C(n) may be defined. These are known to satisfy

k_C(2) in [1.1526,1.2157]
(7)
k_C(3) in [1.2108,1.2744]
(8)
k_C(4) in [1.2413,1.3048]
(9)

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

The limit

k_C=lim_(n->infty)k_C(n)
(10)

satisfies

1.35584631827168<k_C<=1.40491
(11)

(Krivine 1977, 1979; Haagerup 1987; Finch 2003, p. 246; Guo et al. 2026), where the upper limit (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
(12)
=1/x[E(x)-(1-x^2)K(x)],
(13)

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).
(14)

However, Haagerup (1987) has suggested that the upper limit (and presumable actual value) is incorrect and would more plausibly be given by

(int_0^(pi/2)(cos^2theta)/(sqrt(1+sin^2theta))dtheta)^(-1)=1/(2K(i)-E(i))
(15)
=1.4045759...
(16)

(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.

See also

Hilbert Space

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References

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 (4) 043059, 2013.Designolle, S.; Vértesi, T.; and Pokutta, S. "Better Bounds on Grothendieck Constants of Finite Orders." 20 Dec 2024. https://arxiv.org/abs/2409.03739.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." Israeli J. Math. 60, 199-224, 1987.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. Schachermauer). 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 spheres." 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."

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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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