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Five Point Energy Minimization: A Synopsis

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Abstract

This paper is a condensation of my arXiv monograph entitled Schwartz “The phase transition in 5 point energy minimization”, 2016. arXiv:1610.03303, which contains a complete proof that there is a constant such that the triangular bi-pyramid is the minimizer, amongst all 5 point configurations on the sphere, with respect to the power law potential \(R_s(r)=\mathrm{sign}(s)/r^s\), if and only if . In this paper we explain the main ideas and give proofs for some of the key lemmas.

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Fig. 1
Fig. 2

Notes

  1. 1.

    This is the configuration isometric to the one consisting of the north pole, the south pole, and three points placed on the equator in an equilateral triangle.

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Acknowledgements

I would like to thank Ed Saff for suggesting that I write this condensation. I would also like to thank two anonymous referees for some helpful suggestions.

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Correspondence to Richard Evan Schwartz.

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Communicated by Edward B. Saff.

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Schwartz, R.E. Five Point Energy Minimization: A Synopsis. Constr Approx (2020). https://doi.org/10.1007/s00365-020-09500-7

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Keywords

  • Thomson’s problem
  • Energy minimization
  • Computer proof

Mathematics Subject Classification

  • 74G65
  • 65D99