|We present a novel justiﬁcation why k-means clusters should be (hyper)ball-shaped ones. We show that the clusters must be ballshaped 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.|
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