Bounds and Limiting Minimizers for a Family of Interaction Energies

We study a two parameter family of energy minimization problems for interaction energies Eα,β with attractive-repulsive potential Wα,β. We develop a concavity principle, which allows us to provide a lower bound on Eα,β if there exist β₀<β<β₁ with minimizers of Eα,β₀ and Eα,β₁ known. In addition to this, we also derive new conclusions about the limiting behaviour of Eα,β for β≈ 2. Finally, we describe a method to show that, for certain values of (α,β), Eα,β 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 Eα,β for a range of α and β, and an analysis of α,β as a function on parameter space.