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.