Geometric clustering in normed planes

Abstract

Given two sets of points A and B in a normed plane, we prove that there are two linearly separable sets $A^{\prime }$ and $B^{\prime }$ such that $\mathrm{diam}(A^{\prime })\le \mathrm{diam}(A)$, $\mathrm{diam}(B^{\prime })\le \mathrm{diam}(B)$, and $A^{\prime }\cup B^{\prime }=A\cup B$. 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.