Discretizing Manifolds with Minimal Energy

Minimal discrete energy problems arise in a variety of scientific contexts—such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few. Our goal is to analyze the structure of configurations generated by optimal (and near optimal) $N$-point configurations that minimize the Riesz $s$-energy over a bounded surface in Euclidean space. The Riesz $s$-energy potential is simply given by $1/r^s$, where $r$ denotes the distance between pairs of points, and is a generalization of the familiar Coulomb potential. We show how such potentials and their minimizing point configurations are ideal for use in sampling surfaces (and even generating a “near perfect” poppy-seed bagel). Connections to the recent breakthrough results by H. Cohn et al on best-packing and universal optimality in 8 and 24 dimensions will be discussed.