Riesz 2-energy in a disc
Also known asRiesz energy on a discRiesz s-energyminimum energy point configuration
Place n points in a disc of radius 1 so as to minimize the sum of 1/distance² over all pairs.
Formal definition
- ContainerA disc of radius 1 centered at (1, 1), with coordinates ranging from 0 to 2
- SubmissionExactly n distinct points
- ConstraintsEvery point lies inside the container or on its boundary
- EnergyE₂(P) is the Riesz 2-energy of P
- ScoreThe verifier rounds each pair's energy upward to the 10⁻⁹ grid before summing, and displays score(P)/10⁹
- ObjectiveMinimize score(P), which approximates E₂(P) with an upward rounding error of at most 10⁻⁹ per pair
Getting a feel for it
Structural features
Because 1/distance² strongly repels nearby points, small point sets sit on the boundary circle as a regular polygon; as n grows, points enter the interior and form concentric rings. The number of rings and the point count per ring jump at specific values of n.
Known results
For the general theory, see Borodachov, Hardin, and Saff (2019). No published per-n table of optima exists for the disc: n = 9–11 show symmetric constructions and n ≥ 12 show the best configurations found by offline numerical search, all open to challenge. Earlier instances n = 5–8 were retired because a regular n-gon on the boundary already achieves the closed-form energy n(n²−1)/24.
SourceCurrent best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (0) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/riesz-energy-in-a-disc.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/riesz-energy-in-a-disc.json
BibTeX (click to copy)
@misc{minmaxarena-riesz-energy-in-a-disc-2026-10,
title = {{Riesz 2-energy in a disc} (P34)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/riesz-energy-in-a-disc.json},
license = {CC BY 4.0}
}Discussion
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