If a convex polytope in $\mathbb R^n$ has more than $n+1$ vertices, then each point in its interior is a convex combination of the vertices in more than one way. For each interior point, the set of all such convex combinations is itself a convex polytope in a space of dimension equal to the number of vertices of the original polytope minus $n+1$.
Are there any results of interest about the relationship between the original polytope and these other polytopes?