We study a two parameter family of energy minimization problems for interaction energies \(\mathcal{E}_{\alpha,\beta}\) with attractive-repulsive potential \(W_{\alpha,\beta}\) . We develop a concavity principle, which allows us to provide a lower bound on \(\mathcal{E}_{\alpha,\beta}\) if there exist \(\beta_0 < \beta < \beta_1\) with minimizers of \(\mathcal{E}_{\alpha,\beta_0}\) and \(\mathcal{E}_{\alpha,\beta_1}\) known. In addition to this, we also derive new conclusions about the limiting behaviour of \(\mathcal{E}_{\alpha,\beta}\) for \(\beta\approx 2.\) Finally, we describe a method to show that, for certain values of \((\alpha,\beta),\) \(\mathcal{E}_{\alpha,\beta}\) 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 \(\mathcal{E}_{\alpha,\beta}\) for a range of \(\alpha\) and \(\beta\) , and an analysis of \(\inf\mathcal{E}_{\alpha,\beta}\) as a function on parameter space.
