Bounds and Limiting Minimizers for a Family of Interaction Energies

We study a two parameter family of energy minimization problems for interaction energies E, with attractive-repulsive potential W,. We develop a concavity principle, which allows us to provide a lower bound on E, if there exist ₀<<₁ with minimizers of E, 䃐 and E, 䃑 known. In addition to this, we also derive new conclusions about the limiting behaviour of E, for 2. Finally, we describe a method to show that, for certain values of (, ), E, 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 E, for a range of and, and an analysis of, as a function on parameter space.