Fix $n$ and consider the polytope given by the inequalities 
$$x\_i\leq x\_j, \text{ and } 0 \leq x\_i \leq a\_i \text{ for all } 1\leq i<j \leq n,$$
where $a\_i \leq a\_i\leq \dots \leq a\_n$ are fixed positive integers.


Has this family of polytopes been studied anywhere? Do they have a name?
Is there a general formula for its (normalized) volume?



Marked chain polytope, has this been studied?
Set $x\_i=y\_1+\cdots+y\_i$ and $a\_i=x\_1+\cdots+x\_i$. We then we get the so-called "Pitman-Stanley polytope" (see Pitman and Stanley's *[A polytope related to empirical distributions, plane trees, parking functions, and the associahedron](https://arxiv.org/abs/math/9908029))*.


