5 Clusters and Partitions

An important tool needed for the proof of Theorem 2.2 (part (1)) and for other results, moving forward, is to “break” up the set upper E into suitable clusters. This chapter discusses clustering and partitions from various points of view.

5.1 Clusters and Partitions

On an intuitive level, let us suppose that, for example, we are given a n-dimensional compact set upper X embedded in double-struck upper R Superscript n plus 1 and suppose one requires to produce say 10 000 points which “represent” the set upper X. How to do this if the set upper X is described by some geometric property? We may think of a process of “breaking up” a compact set roughly as a “discretization”.

Clustering and partitions of sets upper X subset-of double-struck upper R Superscript n with certain geometry, roughly put, are ways to “discretize” the set and are used in many mathematical subjects for example harmonic ...

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