On the mean radius of permutation polytopes

After calculating the mean value of the support function for the simplex of the coordinate vectors over the unit sphere, we find the mean for a number of polytopes, including the Birkhoff and Asymmetric Traveling Salesman polytopes, and discuss what the latter means for the corresponding Asymmetric Traveling Salesman Problem. We also discuss how to apply these results to efficiently count the vertices of certain other permutation polytopes.