Fix $n$. A convex polytope in $\mathbb{R}^n$ is defined as the nonempty, bounded intersection of finitely many closed half-spaces such that it has nonempty interior.
I'm given the list of vertices of a convex polytope. How can I determine which vertices are connected to which other vertices by edges, faces, ..., facets? I.e. the combinatorial structure of the polytope? I've reduced this problem to determining if a given point is in the convex hull of a list of other points.