Bounds and limiting minimizers for a family of interaction energies

We study a two parameter family of energy minimization problems for interaction energies with attractive-repulsive potential . We develop a concavity principle, which allows us to provide a lower bound on if there exist with minimizers of and known. In addition to this, we also derive new conclusion...

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Bibliographic Details
Published inSampling theory, signal processing, and data analysis Vol. 23; no. 2
Main Author Davies, Cameron
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2025
Subjects
Abstract Harmonic Analysis
Machine Learning
Mathematics
Mathematics and Statistics
Original Article
Signal,Image and Speech Processing
Attractive-repulsive power-law interaction
Convex
Weak convergence of measures
Energy minimization
Aggregation equation
Concave interpolation
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Summary:We study a two parameter family of energy minimization problems for interaction energies with attractive-repulsive potential . We develop a concavity principle, which allows us to provide a lower bound on if there exist with minimizers of and known. In addition to this, we also derive new conclusions about the limiting behaviour of for Finally, we describe a method to show that, for certain values of   cannot be minimized by the uniform distribution over a top-dimensional regular unit simplex. Our results are made possible by two key factors – recent progress in identifying minimizers of for a range of and , and an analysis of as a function on parameter space.
ISSN:2730-5716
2730-5724
DOI:10.1007/s43670-025-00103-6