Complexity of approximating the vertex centroid of a polyhedron

Let be an -polytope in ℝ ^d with vertex set V. The vertex centroid is defined as the average of the vertices in V. We first prove that computing the vertex centroid of an -polytope, or even just checking whether it lies in a given halfspace, are #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an ε distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an -polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to any "sufficiently" non-trivial (for example constant) distance, can be used to construct a fully polynomial-time approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of for any fixed constant δ>0.