Moment methods in energy minimization: New bounds for Riesz minimal energy problems

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We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite-dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We numerically compute the second step of our hierarchy for Riesz s-energy problems with five particles on the two-dimensional unit sphere, where the s = 1 case is the Thomson problem. This yields new numerically sharp bounds (up to high precision) and suggests that the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for five particles on the unit sphere. This is the first time a four-point bound has been computed for a problem in discrete geometry.

Original languageEnglish
Pages (from-to)1407-1453
Number of pages47
JournalAmerican Mathematical Society. Transactions
Issue number2
Publication statusE-pub ahead of print - 2 Oct 2019


  • Four-point bounds
  • Harmonic analysis on spaces of subsets
  • Invariant polynomials
  • Lasserre hierarchy
  • Riesz s-energy
  • Semidefinite programming
  • Thomson problem


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