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).