The Heilbronn Problem for Convex Regions

The following pictures show n points inside a convex region with area 1 so that the area A of the smallest triangle formed by these points is maximized. The smallest area triangles are shown.

3.


A = 1

Trivial.


4.

A = 1/2 = .500

Trivial.


5.

A = (5 - √5) / 10 = .276+

Found by David Cantrell in June 2007.


6.

A = 1/6 = .166+

Found by David Cantrell in June 2007.


7.

A = 1/9 = .111+

Proved by Zhenbing Zeng and Lu Yang in 1995.


8.

A = .0800+

Found by David Cantrell in June 2007.


9.

A = .0640+

Found by David Cantrell in June 2007.


10.

A = .0519+

Found by David Cantrell in June 2007.


11.

A = 2/47 = .0425+

Found by David Cantrell in June 2007.


12.

A = 2/51 = .0392+

Found by David Cantrell in June 2007.


13.

A = .0306+

Found by David Cantrell in June 2007.


14.

A = .0277+

Found by David Cantrell in June 2007.


15.

A = .0244+

Found by David Cantrell in June 2007.


16.

A = .0222+

Found by David Cantrell in June 2007.


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