P32 · Extremal configurations · Formed here

The smallest triangle in a quadrant · n = 11

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 = 11

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

Open the full editor ↗

1y0

1234567891011

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

0.008788461407795329

the smallest triangle's area

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

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"
    ],
    [
      "0.669201131",
      "0.573005062"
    ],
    [
      "0.532932804",
      "0.691082239"
    ],
    [
      "0.354459252",
      "0.716742861"
    ],
    [
      "0.190444565",
      "0.641839864"
    ],
    [
      "0.092962248",
      "0.490154418"
    ],
    [
      "0.092962248",
      "0.309845582"
    ],
    [
      "0.190444565",
      "0.158160136"
    ],
    [
      "0.354459252",
      "0.083257139"
    ],
    [
      "0.532932804",
      "0.108917761"
    ],
    [
      "0.669201131",
      "0.226994938"
    ]
  ]
}

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

Instance parameters

{
  "n": 11
}

The current leader's answer

{
  "points": [
    [
      "0.72",
      "0.4"
    ],
    [
      "0.669201131",
      "0.573005062"
    ],
    [
      "0.532932804",
      "0.691082239"
    ],
    [
      "0.354459252",
      "0.716742861"
    ],
    [
      "0.190444565",
      "0.641839864"
    ],
    [
      "0.092962248",
      "0.490154418"
    ],
    [
      "0.092962248",
      "0.309845582"
    ],
    [
      "0.190444565",
      "0.158160136"
    ],
    [
      "0.354459252",
      "0.083257139"
    ],
    [
      "0.532932804",
      "0.108917761"
    ],
    [
      "0.669201131",
      "0.226994938"
    ]
  ]
}

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