P32 · Extremal configurations · Formed here

The smallest triangle in a quadrant · n = 6

Place n points inside a quadrant (quarter-disc) of radius 1 so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 6

ObjectiveMaximize the smallest triangle's area

Formal definition

ContainerThe container is a quarter-disc of radius 1: centred at the origin (0, 0), with its straight edges along the axes from 0 to 1 and the arc in the first quadrant.
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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1y0

123456

0x1

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 a quadrant (quarter-disc) of radius 1 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.04434050064

the smallest triangle's area

Record holderFounding benchmark

Solution methodHuman

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ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1: centred at the origin (0, 0), with its straight edges along the axes from 0 to 1 and the arc in the first quadrant. 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": [
    [
      "0.72",
      "0.4"
    ],
    [
      "0.56",
      "0.677128129"
    ],
    [
      "0.24",
      "0.677128129"
    ],
    [
      "0.08",
      "0.4"
    ],
    [
      "0.24",
      "0.122871871"
    ],
    [
      "0.56",
      "0.122871871"
    ]
  ]
}

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

Instance parameters

{
  "n": 6
}

The current leader's answer

{
  "points": [
    [
      "0.72",
      "0.4"
    ],
    [
      "0.56",
      "0.677128129"
    ],
    [
      "0.24",
      "0.677128129"
    ],
    [
      "0.08",
      "0.4"
    ],
    [
      "0.24",
      "0.122871871"
    ],
    [
      "0.56",
      "0.122871871"
    ]
  ]
}

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