An elementary and unified proof of Grothendieck’s inequality

Abstract

We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current bestknown explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss-Pe lczyński, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.2010 Mathematics Subject Classification. 47A07, 46B28, 46B85, 81P40, 81P45, 03D15, 97K30, 47N10, 90C27.

Two inequalities: a geometric and a combinatorial

“…This proves (3). The first inequality in (2) has on both sides fractions of symmetric functions in a, b, c. By expressing numerators and denominators in terms of e 1 , e 2 and e 3 , it is not hard to show that this is equivalent to the inequality (1). Finally, since the equality R/r = 2 holds if and only if the triangle is equilateral, it follows that equality in (1) holds if and only if a = b = c. We can write (2) also in the form…”

“…Perhaps a good starting point to think about such general problems is Grothendieck's inequality (see e.g. [1]).…”

Grothendieck's inequality and completely correlation preserving functions -- a summary of recent results and an indication of related research problems

“…A slightly bit modified version of this rewritten version of Grothendieck's inequality reads as follows (cf. [13,Lemma 2.2]):…”