Birkhoff polytope

The Birkhoff polytope Bn 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 n2-dimensional Euclidean space, the entries are unraveled in reading order. Bn is (n - 1)2-dimensional, as it lies within an affine subspace of that dimension.

Birkhoff's theorem states that the vertices of Bn 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!. B2 is a line segment, while for n > 2 Bn has n2 facets. Bn is centered on the point corresponding to the n &times; n matrix all of whose entries are 1/n.

The four-dimensional B3 has six vertices, which are placed in two congruent equilateral triangles in orthogonal planes. It has 15 edges. The two equilateral triangles total six edges of side length $$\sqrt{6}$$, while the other nine have side length 2.

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