Generalized permutahedron and random polytopes 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 $(1, \ldots, n)$.
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?  
 A: As long as the $a_i$ are all distinct, the resulting polytope is combinatorially equivalent to the "standard permutahedron". So you will obtain the essentially same polytope with probability one (for reasonable probability distributions). See for example here (their definition of "generalized permutahedron" is distinct from yours, but what you call "generalized permutahedron" for distinct $a_i$ is there just called "permutahedron"). To get an idea for the proof of this fact, consider this answer, in which I derive the edge structure of the standard permutahedron. The argument does not really depend on the fact that we use the vector $(1,...,n)$, but just that the entries are distinct. 
The other polytopes you can obtain in this way (when the $a_i$ are not neccessarily distinct) are the orbit polytopes of the $A_n$ reflection group. Each of these is combinatorially equivalent to one of the uniform polytope from the $A_n$-family (see e.g. here for dimension four, and there are pages for the other dimensions too). 
