The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the permutahedron, which is defined as the convex hull of the set of vertices obtained by permuting the entries of the vector $(1,…,n)$.

An obvious generalization of the permutahedron is to consider the convex hull of all the vectors that can obtained by permuting the entries of an arbitrary vector $(a_1, \ldots, a_n) \in \mathbb{R}^n$. I would like pointers to information about the generalized permutahedron.

What if the entries of $(a_1,\ldots,a_n)$ are sampled according to a random distribution (e.g, gaussian with zero mean and some variance). This yields a random polytope. What can be said about it?