Addendum to Minimal Riesz Energy Point Configurations for Rectifiable D-dimensional Manifolds *

Abstract

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/rs, where s > 0 and r is Euclidean distance in Rd ′ . With Es(A,N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A,N) for each fixed s ≥ d. Moreover, if A has positive d-dimensional Hausdorff measure, we show that N -point configurations on A that minimize the s-energy are asymptotically uniformly distributed with respect to d-dimensional Hausdorff measure on A when s ≥ d. Even for the unit sphere Sd ⊂ Rd+1, these results are new.

Cite this paper

@inproceedings{Hardin2008AddendumTM, title={Addendum to Minimal Riesz Energy Point Configurations for Rectifiable D-dimensional Manifolds *}, author={Doug P. Hardin and Edward B. Saff}, year={2008} }