for all real numbers , , ..., and , , ..., such that . Then Grothendieck showed that there exists
a constant
satisfying
(2)
for all vectors
and
in a Hilbert space with norms and . The Grothendieck constant is the smallest possible
value of .
For example, the best values known for small are
(3)
(4)
(5)
(Krivine 1977, 1979; König 1992; Finch 2003, p. 236).
Now consider the limit
(6)
which is related to Khinchin's constant and sometimes also denoted . Krivine (1977) showed that
(7)
and postulated that
(8)
(OEIS A088367). The conjecture was refuted in 2011 by Yury Makarychev, Mark Braverman, Konstantin Makarychev, and Assaf Naor,
who showed that
is strictly less than Krivine's bound (Makarychev 2011).
Similarly, if the numbers and and matrix are taken as complex, then a similar set of constants
may be defined. These are known to satisfy
(9)
(10)
(11)
(Krivine 1977, 1979; König 1990, 1992; Finch 2003, p. 236).
The limit
(12)
satisfies
(13)
(Krivine 1977, 1979; Haagerup 1987; Finch 20003, p. 246), where the upper limit (OEIS A088374) is given by with
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.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 -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 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."