The Extreme Points of Fusions

TL;DR

This paper investigates the geometric structure of fusions, a multidimensional generalization of mean-preserving contractions, revealing their connection to power diagrams and their role in natural and categorization processes.

Contribution

It introduces a geometric and combinatorial framework for understanding fusions and links Lipschitz-exposed points to power diagrams, advancing the theoretical understanding of these objects.

Findings

Identified the structure of extreme and exposed points of fusions.

Connected Lipschitz-exposed points to power diagrams.

Applied the framework to categorization problems.

Abstract

Our work explores fusions, the multidimensional counterparts of mean-preserving contractions and their extreme and exposed points. We reveal an elegant geometric/combinatorial structure for these objects. Of particular note is the connection between Lipschitz-exposed points (measures that are unique optimizers of Lipschitz-continuous objectives) and power diagrams, which are divisions of a space into convex polyhedral ``cells'' according to a weighted proximity criterion. These objects are frequently seen in nature--in cell structures in biological systems, crystal and plant growth patterns, and territorial division in animal habitats--and, as we show, provide the essential structure of Lipschitz-exposed fusions. We apply our results to several questions concerning categorization.