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

Published:2011-09-15
Printed: Feb 2012
• S. V. Borodachov,
Department of Mathematics, Towson University, 8000 York Road, Towson, MD, 21252, USA
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## Abstract

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in $\mathbb{R}^d$ with the potential $|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ 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 $s\geq 2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length of the curve.
 Keywords: minimal discrete Riesz energy, lower order term, power law potential, separation radius
 MSC Classifications: 31C20 - Discrete potential theory and numerical methods 65D17 - Computer aided design (modeling of curves and surfaces) [See also 68U07]

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