Grothendieck's Constant

Let  be an  real square matrix with  such that

(1)

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

			(14)
			(15)

 a complete elliptic integral of the second kind,  a complete elliptic integral of the first kind, and  (OEIS A088373) the root of

(16)

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

			(17)
			(18)

(OEIS A088375; Finch 2003, pp. 236-237).

REFERENCES:

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."