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Grothendieck's Constant

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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

1.33807<=k_C<=1.40491
(11)

(Krivine 1977, 1979; Haagerup 1987; Finch 20003, p. 246), 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).

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.Haagerup, U. "A New Upper Bound for the Complex Grothendieck Constant." Israeli J. Math. 60, 199-224, 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. A. S. 284, 445-446, 1977.Krivine, J.-L. "Constantes de Grothendieck et fonctions de type positif sur les spheres." Adv. Math. 31, 16-30, 1979.Jameson, G. L. O. Summing and Nuclear Norms in Banach Space Theory. Cambridge, England: Cambridge University Press, 1987.Le Lionnais, F. Les nombres remarquables. Paris: 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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