MinMax Arena
中/EN
P34 · Extremal configurations · Classic · Weak baseline

Riesz 2-energy in a disc · n = 5

Place n points inside a circle of radius 1, minimizing the sum of 1/distance² taken over every pair.

Instancen = 5
ObjectiveMinimize Riesz 2-energy
Best known, unproven5regular pentagon vertices, E = n(n²−1)/24 = 5

Formal definition

  • ContainerThe container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2.
  • SubmissionExactly n points, no two coinciding
  • ConstraintsEvery point lies inside the container or on its boundary
  • ObjectiveMake the sum of 1/distance² over all pairs as small as possible; scored in exact rationals
Open the full editor ↗
2y0
0x2
VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Where the room for improvement is

1/distance² punishes closeness brutally: points get pushed out to a boundary ring first, then shed inner layers as n grows. Layer counts jump at particular n, and the jumps are where the contest lives.

Where the frontier is

Same theory as the square version: Borodachov, Hardin and Saff (2019); no per-n table of optima in a disc either. The regular-polygon closed forms are trivial cases proved on site; everything else is open.

Source
Current leader

5.000000008

Riesz 2-energy

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (3 changes)
  1. AnonymousHuman
    5.520154215→5.000000008
  2. SeiaHuman
    182.64574973→5.520154215
  3. SeiaHuman
    250.200002011→182.64574973
ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.5". The score is the sum of 1/distance² over every pair, and smaller is better.

The current leader's answer

{
  "points": [
    [
      "1.951056516",
      "1.309016994"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.048943484",
      "1.309016994"
    ],
    [
      "0.412214748",
      "0.190983006"
    ],
    [
      "1.587785252",
      "0.190983006"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 5
}

The current leader's answer

{
  "points": [
    [
      "1.951056516",
      "1.309016994"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.048943484",
      "1.309016994"
    ],
    [
      "0.412214748",
      "0.190983006"
    ],
    [
      "1.587785252",
      "0.190983006"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". The score is the sum of 1/distance² over every pair, and smaller is better. · Verifier v1.0.0