P28 · Extremal configurations · Formed here

The smallest triangle in an annulus · n = 8

Place n points inside an annulus with outer radius 1 and inner radius 0.5 so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 8

ObjectiveMaximize the smallest triangle's area

Formal definition

ContainerThe container is an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it.
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest triangle over all triples as large as possible; compared internally by twice the area, exactly

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2y0

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0x2

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Where the room for improvement is

Even spreading is not enough: no three points may come close to collinear, and near-collinearity is exactly what tidy arrangements love to do. Optima are often asymmetric and hard even to guess.

Where the frontier is

Our own variant: Heilbronn's problem in an annulus with outer radius 1 and inner radius 0.5 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.0082842712

the smallest triangle's area

Record holderFounding benchmark

Solution methodHuman

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ANSWER FORMAT

How to write your answer

The container is an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it. 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 current leader's answer

{
  "points": [
    [
      "1.95",
      "1"
    ],
    [
      "1.891421356",
      "1.141421356"
    ],
    [
      "1.75",
      "1.2"
    ],
    [
      "1.608578644",
      "1.141421356"
    ],
    [
      "1.55",
      "1"
    ],
    [
      "1.608578644",
      "0.858578644"
    ],
    [
      "1.75",
      "0.8"
    ],
    [
      "1.891421356",
      "0.858578644"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "points": [
    [
      "1.95",
      "1"
    ],
    [
      "1.891421356",
      "1.141421356"
    ],
    [
      "1.75",
      "1.2"
    ],
    [
      "1.608578644",
      "1.141421356"
    ],
    [
      "1.55",
      "1"
    ],
    [
      "1.608578644",
      "0.858578644"
    ],
    [
      "1.75",
      "0.8"
    ],
    [
      "1.891421356",
      "0.858578644"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". · Verifier v1.0.0