Abstract
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. As a result, for a given k, 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 is studied when two different bounds are imposed to the diameters. The Hershberger–Suri's data structure for managing ball hulls can be useful in this context.
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