Question

Birkhoff's theorem states:

The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices

This theorem seems to be commonly attributed to Birkhoff (perhaps also von Neumann). But I recall listening to a talk by Harold Kuhn, where he said that this theorem should actually be attributed to some $P$ where $P \in \{$Jacobi, Dénes Kőnig, Jenő Egerváry, Somebody else?$\}$.

Question: Does anybody know whom Kuhn might have meant, and to whom this theorem should really be attributed?

I would be very happy to learn the connection (also, yes, am embarrassed that despite listening carefully during the talk, I have still forgotten!)

Answer 1

See the Wikipedia page for the Birkhoff polytope. It says that equivalent results were obtained by Steinitz in 1894 and by Kőnig in 1916.

Answer 2

From Schneider's "The Birkhoff-Egervary-Konig theorem for matrices over lattice ordered abelian groups" after the following theorem is presented

Let $G$ be a lattice ordered abelian group. Every generalized doubly stochastic matrix with elements from $G$ is the sum of generalized permutation matrices.

it is remarked that the theorem for $G=\mathbb{Z}$ was obtained by Konig in 1916 and for $G=\mathbb{R}$ by Birkhoff in 1946 however in 1931 "Egervary proved a result for integral matrices which is more general than Konig's theorem. He observed that by continuity considerations his theorem may be shown to hold for real matrices. Thus in this way one obtains a result which contains Birkhoff's theorem." There are many references in the paper.