Combining the ideas of Riesz s-energy and log-energy, we introduce the so-called s,log^t-energy. In this research project, we investigate the asymptotic behaviors of minimal N-point s,log^t-energy constants and configurations of an infinite compact metric space of diameter less than 1 when the variables N,t are fixed but the variable s is varied. In particular, we study certain continuity and differentiability properties of minimal N-point s,log^t-energy constants with respect variable s and we show that in the limits as s→∞ and as s→s_0>0, minimal N-point s,log^t-energy configurations tend to an N-point best-packing configuration and a minimal N-point s_0,log^t-energy configuration, respectively. Furthermore, the optimality of N distinct equally spaced points on circles in R^2 for some certain s,log^t energy problems was proved.

Reference

N. Loesatapornpipit, N. Bosuwan, Asymptotic Properties of Discrete Minimal s,log^t-Energy Constants and Configurations. Symmetry. 2021, 13, 932. https://doi.org/10.3390/sym13060932