Some clustering algorithms 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 $\mathrm{diam}\left(A^{\prime }\right)\le \mathrm{diam}\left(A\right),\mathrm{diam}\left(B^{\prime }\right)\le \mathrm{diam}\left(B\right)$, and $A^{\prime }\cup B^{\prime }=A\cup B$. As a result, some Euclidean clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the diameters (or the radii) of k clusters. Some specific solutions are presented for $k=2$ and $k=3$ that minimize the diameter of the clusters. The 2-clustering problem when two different bounds are imposed to the diameters is also studied.