TITLE: Generalized permutahedron and random polytopes
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: If you're asking about the combinatorics of the polytope, then there is an easy answer.  If you sample with a continuous distribution, then you will get a combinatorial permutohedron with probability 1.  The only way to get something else is to take a point on one or more of the hyperplanes x_i=x_j.  Then your orbit (i.e. set of vertices of the polytope) gets smaller.  The simplest way to understand the number of vertices is to think about the stabilizer of your point:  It is the subgroup generated by all transpositions (i j) such that your initial vector has x_i=x_j.  So it's a product of symmetric groups.
Also, I agree with Benjamin Steinberg (comment above) that you want to look at the Postnikov reference.