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Extended convexity and uniqueness of minimizers for interaction energies

Ruiwen Shu

TL;DR

This work extends the LIC framework for the uniqueness of minimizers of interaction energies by introducing and quantifying an LIC radius, $R_{LIC}[W]$, that can exceed a uniform bound on minimizer support $R_*$. Leveraging a Fourier-based representation of the energy and sharp Poincaré-type inequalities for signed measures, the authors prove uniqueness (up to translations) for power-law potentials $W_{a,b}$ in new regimes: $a$ near 2 or near 4 for any $-d<b<2$, and for the logarithmic power-law $W_{b,\ln}$ near $b=2$. For $a>4$, they implement a truncation strategy to produce a positive Fourier transform for a truncated potential, ensuring a lower bound on the LIC radius and hence uniqueness. The results connect explicit minimizer profiles (including density formulas and concentration on spheres) with rigorous convexity and Fourier-analytic criteria, broadening the class of interaction energies for which uniqueness can be certified. Overall, the paper provides a robust framework to deduce uniqueness from extended convexity via LIC radius, with quantitative tools that may guide future analyses of non-LIC potentials.

Abstract

Linear interpolation convexity (LIC) has served as the crucial condition for the uniqueness of interaction energy minimizers. We introduce the concept of the LIC radius which extends the LIC condition. Uniqueness of minimizer up to translation can still be guaranteed if the LIC radius is larger than the possible support size of any minimizer. Using this approach, we obtain uniqueness of minimizer for power-law potentials $W_{a,b}({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b},\,-d<b<2$ with $a$ slightly smaller than 2 or slightly larger than 4. The estimate of LIC radius for $a$ slightly smaller than 2 is done via a Poincaré-type inequality for signed measures. To handle the case where $a$ slightly larger than 4, we truncate the attractive part of the potential at large radius and prove that the resulting potential has positive Fourier transform. We also propose to study the logarithmic power-law potential $W_{b,\ln}({\bf x}) = \frac{|{\bf x}|^b}{b}\ln|{\bf x}|$. We prove its LIC property for $b=2$ and give the explicit formula for minimizer. We also prove the uniqueness of minimizer for $b$ slightly less than 2 by estimating its LIC radius.

Extended convexity and uniqueness of minimizers for interaction energies

TL;DR

This work extends the LIC framework for the uniqueness of minimizers of interaction energies by introducing and quantifying an LIC radius, , that can exceed a uniform bound on minimizer support . Leveraging a Fourier-based representation of the energy and sharp Poincaré-type inequalities for signed measures, the authors prove uniqueness (up to translations) for power-law potentials in new regimes: near 2 or near 4 for any , and for the logarithmic power-law near . For , they implement a truncation strategy to produce a positive Fourier transform for a truncated potential, ensuring a lower bound on the LIC radius and hence uniqueness. The results connect explicit minimizer profiles (including density formulas and concentration on spheres) with rigorous convexity and Fourier-analytic criteria, broadening the class of interaction energies for which uniqueness can be certified. Overall, the paper provides a robust framework to deduce uniqueness from extended convexity via LIC radius, with quantitative tools that may guide future analyses of non-LIC potentials.

Abstract

Linear interpolation convexity (LIC) has served as the crucial condition for the uniqueness of interaction energy minimizers. We introduce the concept of the LIC radius which extends the LIC condition. Uniqueness of minimizer up to translation can still be guaranteed if the LIC radius is larger than the possible support size of any minimizer. Using this approach, we obtain uniqueness of minimizer for power-law potentials with slightly smaller than 2 or slightly larger than 4. The estimate of LIC radius for slightly smaller than 2 is done via a Poincaré-type inequality for signed measures. To handle the case where slightly larger than 4, we truncate the attractive part of the potential at large radius and prove that the resulting potential has positive Fourier transform. We also propose to study the logarithmic power-law potential . We prove its LIC property for and give the explicit formula for minimizer. We also prove the uniqueness of minimizer for slightly less than 2 by estimating its LIC radius.

Paper Structure

This paper contains 19 sections, 23 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. The LIC radius and related concepts
  3. Estimating the LIC radius
  4. Some discontinuous behavior of the LIC radius
  5. Organization of this paper
  6. Preparations
  7. The Euler-Lagrange condition
  8. LIC radius and the Euler-Lagrange condition
  9. Uniform estimate on the size of minimizers
  10. Fourier representation of the interaction energy
  11. Fourier representable potentials at level 0
  12. Fourier representable potentials at levels 1 and 2
  13. A dimension reduction trick
  14. Application to power-law potentials
  15. Fourier representation of the LIC radius
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume (W0). Assume $\lim_{|{\bf x}|\rightarrow\infty}W({\bf x})=:W_\infty$ is a real number or $\infty$. If there exists $\rho\in \mathcal{M}(\mathbb{R}^d)$ such that $E[\rho]< \frac{1}{2}W_\infty$, i.e., then there exists a minimizer of $E$.

Theorems & Definitions (55)

  • Theorem 1.1: SST15
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Lemma 1.8
  • proof
  • Remark 2.1
  • ...and 45 more