TITLE: Pictures of the von Neumann polytope
QUESTION [9 upvotes]: Are there any graphic portrayals of von Neumann polytopes in low dimensions?

REPLY [3 votes]: The set of doubly stochastic matrices is a subset of the set of
nonnegative $n \times n$ matrices whose sum of entries is $n$. 
These matrices form an $(n^2-1)$-dimensional simplex in $\mathbb{R}^{n^2}$, whose vertices are the $n^2$ matrices (written as row vectors of length $n^2$) $(n,0,0,...,0)$, $(0,n,0,...,0)$, ..., $(0,0,... 0,n)$.
The vertices of the Birkhoff polytope are the arithmetic means of $n$ of these - suitably chosen - matrices.
For $n=2$ the Birkhoff polytope lies within a tetrahedron, between the midpoints of the edges that connect points $1 = (2, 0, 0, 0)$ and $4 = (0, 0, 0, 2)$ and points $2 = (0, 2, 0, 0)$ and $3 = (0, 0, 2, 0)$.