The Extreme Points of Fusions

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.