I cannot understand a few things about the Birkhoff polytope. The Birkhoff polytope is defined as a polyhedron of all $n \times n$ doubly stochastic matrices. How is is it polytope then? To transform $n \times n$ matrix into a point in $n$-dimensional Euclidean space, we should multiply it by a fixed vector or something? I couldnt find any explanaitions about how it does work. Also, does someone have full proof of the fact that in Birkhoff polytope extreme points (vertices) are exactly permutation matrices?
I can't also fully understand the difference beetwen Birkhoff polytope and permutation polyhedron (permutohedron). The second one is defined as convex hull of all permutations of fixed point, at it is deeply connected to Birkhoff polytope but I don't quite see the difference, aren't these exactly the same thing?