Optimal partitions

Abstract

This chapter provides a survey on the problem of partitioning points in the d-dimensional space with the goal of maximizing a given objective function; the survey is based on the forthcoming book (Hwang FK, Rothblum UG (2011b) Partitions: Optimality and clustering. World Scientific). The first two sections describe the framework, terminology, and some background material that are useful for the analysis of the partition problem. The middle two sections study the case of d = 1, and the last two sections study the case of general d. Results are presented without proofs, but with motivations, comments, and references. Complete proofs can also be found in the forthcoming book.