2017/09/13 by Martín, Pedro, Yáñez, Diego
#46B20 #52A10 #52A21 #52B55 #65D18 #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1709.04976
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. 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.