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 $A'$ and $B'$ such that $\mathrm{diam}(A')\leq \mathrm{diam}(A)$, $\mathrm{diam}(B')\leq \mathrm{diam}(B)$, and $A'\cup B'=A\cup B.$ 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.