Einstein defined gravity as a curvature of spacetime. Therefore, a sphere presents an easily definable center of mass.

How exactly would the elevator be set up in your thought experiment? If you’re considering the standard elevator, within a building on Earth, then the distance the of the objects’ shared fall would be so minuscule in comparison to their mean distance from the gravitational warp of the earth’s center that any movement within the elevator toward a common point would be virtually undetectable.

• GR postulates local equivalency between gravity and acceleration - alexander - October 2, 2001 - 00:23 UTC