In this talk, we explore optimal point configurations in $R^d$, emphasizing the case where our kernel is periodized by a lattice. We then present a few new results on the optimality of certain low-cardinality configurations for families of completely monotone potentials periodized by the hexagonal lattice and an associated sublattice. Our main tools are the Delsarte-Yudin linear programming method along with some basic facts regarding polynomial interpolation. We also outline some conjectures and future plans to extend our results to higher cardinality configurations. All work is joint with Doug Hardin.