Let us fix $\Delta$ a finite simplicial complex, and label the vertices of $\Delta$ as $\{1,2,\ldots,n\}$. For each $F\in \Delta$ let us consider the point in $\mathbb{R}^n$ given by:

$$e_F := \sum_{i\in F} e_i,$$

where $e_i$ denotes the $i$-th canonical vector in $\mathbb{R}^n$. Let us define the polytope $\mathscr{P}(\Delta)$ as the convex hull in $\mathbb{R}^n$ of all these $e_F$ for $F$ ranging in $\Delta$.

Why is this construction interesting? Well, it contains two famous families of polyhedra:

- Independence matroid polytopes. This is easy to see, since the family of independent sets of a matroid is indeed a simplicial complex.
- Chain polytopes of posets. This is a little bit harder, but still elementary: the vertices of the chain polytope of a poset $P$ are the antichains of $P$, and this defines a simplicial complex.

I have two questions. Has this family of polytopes already been studied? Are there any other big (or interesting) family of polytopes that fall into this category?

As a side note: there is a nice formula for the volume of chain polytopes, I do not know if this is too naive, but maybe this could help to guess a "good" formula for the volume of independence matroid polytopes.