We present a novel justification why k-means clusters should be (hyper)ball-shaped ones. We show that the clusters must be ball-shaped to attain motion-consistency. If clusters are ball-shaped, one can derive conditions under which two clusters attain the global optimum of k-means. We show further that if the gap is sufficient for perfect separation, then an incremental k-means is able to discover perfectly separated clusters. This is in conflict with the impression left by an earlier publication by Ackerman and Dasgupta. The proposed motion-transformations can be used to the new labeled data for clustering from existent ones.