altitude data?

This entire discussion is so ridiculous that I'm having trouble believing it isn't all a joke, but surely the answer is that, assuming the Earth to be an oblate spheroid of constant density (not actually true but apparently close enough), any point lying on the equator, plus the poles, would be 'directly above' the centre of mass, if by that you mean that the surface normal at that point intersects the centre (NB. this is based on my intuitive geometric understanding of the spheroid. I haven't attempted to verify it mathematically, though doing so should be straightforward).

If you actually care about the Euclidean distance from the centre, then clearly the poles are going to be the closest points, and given that the density of the Earth isn't really constant, one is likely to be closer than the other (an earlier poster claimed that the north pole is closer to the Earth's centre of mass than the south). Note that the equatorial region, though 'directly above' the centre of mass, is furthest from it.