Birkhoff's Inequality

In homogeneous coordinates, the first positive quadrant joins  with  by "points" , and is mapped onto the hyperbolic line  by the correspondence . Now define

(1)

Let  be any bounded linear transformation of a Banach space  that maps a closed convex cone  of  onto itself. Then the -norm  of  is defined by

(2)

for pairs  with finite . Birkhoff's inequality then states that if the transform  of  under  has finite diameter  under , then

(3)

(Birkhoff 1957).

REFERENCES:

Birkhoff, G. "Extensions of Jentzsch's Theorem." Trans. Amer. Math. Soc. 85, 219-227, 1957.

Jentzsch, R. "Über Integralgleichungen mit positivem Kern." J. reine angew. Math. 141, 235-244, 1912.

Schmeidler, W. Integralgleichungen mit Anwendungen in Physik und Technik, Vol. 1. Lineare Integralgleichungen. Leipzig, Germany: Geest & Portig, p. 298, 1955.