Birkhoff polytope

The Birkhoff polytope ${\displaystyle B_{n}}$ is the convex polytope in ${\displaystyle \mathbb {R} ^{n^{2}}}$ whose points are the doubly stochastic matrices, that is, the ${\displaystyle n\times n}$ matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff, and also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph ${\displaystyle K_{n,n}}$.