[1]) is the convex polytope in RN (where N = n2) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1.
[1] This follows from the Birkhoff–von Neumann theorem, which states that the extreme points of the Birkhoff polytope are the permutation matrices, and therefore that any doubly stochastic matrix may be represented as a convex combination of permutation matrices; this was stated in a 1946 paper by Garrett Birkhoff,[2] but equivalent results in the languages of projective configurations and of regular bipartite graph matchings, respectively, were shown much earlier in 1894 in Ernst Steinitz's thesis and in 1916 by Dénes Kőnig.
The edges of the Birkhoff polytope correspond to pairs of permutations differing by a cycle: This implies that the graph of Bn is a Cayley graph of the symmetric group Sn.
The Birkhoff polytope lies within an (n2 − 2n + 1)-dimensional affine subspace of the n2-dimensional space of all n × n matrices: this subspace is determined by the linear equality constraints that the sum of each row and of each column be one.
[6] The following asymptotic formula was found by Rodney Canfield and Brendan McKay:[7] For small values
[9] Determining the Ehrhart polynomial of a polytope is harder than determining its volume, since the volume can easily be computed from the leading coefficient of the Ehrhart polynomial.