Birkhoff polytope

The Birkhoff polytope $B_{n}$ is a convex polytope whose points are the doubly stochastic $n &times; n$ matrices. A matrix is called doubly stochastic if its rows and columns each sum to 1. To convert matrices to points in $n^{2}$-dimensional Euclidean space, the entries are unraveled in reading order. $B_{n}$ is $(n - 1)^{2}$-dimensional, as it lies within an affine subspace of that dimension.

Birkhoff's theorem states that the vertices of $B_{n}$ are the $n &times; n$ permutation matrices (defined as matrices where each row and column contains exactly one 1 and all other entries are 0), of which there are $n!$. $B_{2}$ is a line segment, while for $n > 2$, $B_{n}$ has $n^{2}$ facets. $B_{n}$ is centered on the point corresponding to the $n &times; n$ matrix all of whose entries are $1/n$.

The four-dimensional $B_{3}$ is a triangular duotegum with edge lengths 2 and $$\sqrt{6}$$.

Birkhoff polytopes are isogonal and isotopic, and are therefore noble polytopes.