The Convex Hull of Parking Functions of Length $n$

Let ${\mathcal{P}}_n$ be the convex hull in ${\mathbb{R}}^n$ of all parking functions of length $n$. Stanley found the number of vertices and the number of facets of ${\mathcal{P}}_n$. Building upon these results, we determine the number of faces of arbitrary dimension, the volume, and the number of integer points of ${\mathcal{P}}_n$.