How to find the center of mass (not vertex average) of a convex hull?

Question

I have found results that say that computing the average of vertices of a polytope presented by inequalities is a #$P$-hard problem. However what if we want the true center of mass (determined by volumes) and we are given vertices instead of inequalities? Is there any fast way to get the center of mass? If not, is there a proof I haven't found that the problem is hard in some sense, unless we fix the dimension of the polytope? I know the result that if the dimension is fixed then we can triangulate the polytope and get the center of mass in polynomial time.

Answer 1

This might be interesting for you: 'Approximating the centroid is hard' (L. Rademacher) http://web.cse.ohio-state.edu/~lrademac/centroid.pdf

As far as I know the most common way to compute the center of mass of a polytope would be to triangulate it into simplices. Then the center of mass of a simplex coincides with its vertex average.

Answer 2

One way to do this is to divide the convex hull into non-overlapping tetrahedra, find the center (mean of the 4 vertices) and volume of each tetrahedra and average all the centers weighted by the volume.

Since the polytope is convex, we can create non-overlapping tetrahedra by selecting each triangle on the surface and connecting it with the center point (by averaging all vertices).