2024/09/16 by Andreas Kleiner, Kleiner, Andreas, Benny Moldovanu +5 · 3 citations
Physics and Astronomy · #FOS: Economics and business #FOS: Mathematics #Probability (math.PR) #Space Science and Extraterrestrial Life #Theoretical Economics (econ.TH)
paper · pdf · doi:10.48550/arxiv.2409.10779
openalex publication_date 2024/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.