Geometric clustering in normed planes

TL;DR

This paper extends Euclidean clustering results to normed planes, demonstrating that point sets can be partitioned into linearly separable subsets with controlled diameters, enabling adaptation of clustering algorithms to various normed spaces.

Contribution

It generalizes a Euclidean clustering partitioning result to symmetric convex normed planes and adapts Euclidean clustering algorithms for these spaces.

Findings

Partitioning of point sets into linearly separable subsets with controlled diameters.

Extension of Euclidean clustering algorithms to normed planes.

Solution to 2-clustering with different diameter bounds.

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.