Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Abstract

AbstractWe consider the problem of minimizing the energy of N points repelling each other on curves in ℝ ∞d with the potential |x — y|—s, s ≥ 1, where |・| is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for s ≥ 2, the minimal pairwise distance in optimal configurations asymptotically equals L/N, N → 1, where L is the length of the curve.