# 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 $(1,…,n)$.

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?

• I think people generally use permutohedron to refer to the convex hull of all permutations of the entries of any vector with distinct entries. If the entries are distinct you always get the same face lattice, which is just the dual to the face lattice of the braid hyperplane arrangement. Maybe look at www-math.mit.edu/~apost/papers/permutohedron.pdf – Benjamin Steinberg Apr 9 '14 at 17:28
• @rnegrinho This has been inactive for a while. Did you find anything else? Thanks. – RMurphy Jul 15 '19 at 0:08