For \({\textbf {b}}=(b_1,\dots ,b_n)\in \mathbb {Z}_{>0}^n\) , a \({\textbf {b}}\) -parking function is defined to be a sequence \((\beta _1,\dots ,\beta _n)\) of positive integers whose nondecreasing rearrangement \(\beta '_1\le \beta '_2\le \cdots \le \beta '_n\) satisfies \(\beta '_i\le b_1+\cdots + b_i\) . The \({\textbf {b}}\) -parking-function polytope \(\mathfrak {X}_{n}({\textbf {b}})\) is the convex hull of all \({\textbf {b}}\) -parking functions of length n in \(\mathbb {R}^n\) . Geometric properties of \(\mathfrak {X}_{n}({\textbf {b}})\) were previously explored in the specific case where \({\textbf {b}}=(a,b,b,\dots ,b)\) and were shown to generalize those of the classical parking-function polytope. In this work, we study \(\mathfrak {X}_{n}({\textbf {b}})\) in full generality. We present a minimal inequality and vertex description for \(\mathfrak {X}_{n}({\textbf {b}})\) , prove it is a generalized permutahedron, and study its h-polynomial. Furthermore, we investigate \(\mathfrak {X}_{n}({\textbf {b}})\) through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.
