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A simple and e cient algorithm for nding the closest points between two convex polyhedra is described here. Data from numerous experiments tested on a broad set of convex polyhedra on< 3 show that the running time is roughly constant for nding closest points when nearest points are approximately known and is linear in total number of vertices if no special initialization is done. This algorithm can be used for collision detection, computation of the distance between two polyhedra in three-dimensional space, and other robotics problems. It forms the heart of the motion planning algorithm of 1].