Geometric clustering in normed planes
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Given two sets of points A and B in a normed plane, we prove that there are two linearly separable sets A' and B' such that diam(A')≤diam(A), diam(B')≤diam(B), and A'∪ B'=A∪ B. This extends a result for the Euclidean distance to symmetric convex distance functions. As a consequence, some Euclidean k-clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the k cluster diameters. The 2-clustering problem when two different bounds are imposed to the diameters is also solved. The Hershberger-Suri's data structure for managing ball hulls can be useful in this context.
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