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Summary: Minkowski-Type Theorems and Least-Squares Clustering
Franz Aurenhammer
Institute for Theoretical Computer Science
Graz University of Technology
Graz, Austria
Friedrich Ho mann
Institut fur Informatik
Technische Universitat Munchen
Munchen, Germany
Boris Aronov
Computer and Information Science Department
Polytechnic University
New York, USA
Abstract
Dissecting Euclidean d-space with the power diagram of n weighted point sites partitions
a given m-point set into clusters, one cluster for each region of the diagram. In this manner,
an assignment of points to sites is induced. We show the equivalence of such assignments to
constrained Euclidean least-squares assignments. As a corollary, there always exists a power
diagram whose regions partition a given d-dimensional m-point set into clusters of prescribed
sizes, no matter where the sites are placed. Another consequence is that constrained least-
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