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.

3-fold dihedral symmetry.

4. A = 1/2 = .500

Trivial.

4-fold dihedral symmetry.

5. A = (5 – √5) / 10 = .276+

Found by David Cantrell in June 2007.

5-fold dihedral symmetry.

6. A = 1/6 = .166+

Found by David Cantrell in June 2007.

6-fold dihedral symmetry.

7. A = 1/9 = .111+

Proved by Zhenbing Zeng and Lu Yang in 1995.

3-fold dihedral symmetry.

8. A = .0800+

Found by David Cantrell in June 2007.

2-fold symmetry.

9. A = .0640+

Found by David Cantrell in June 2007.

2-fold symmetry.

10. A = .0519+

Found by David Cantrell in June 2007.

2-fold dihedral symmetry.

11. A = 2/47 = .0425+

Found by David Cantrell in June 2007.

2-fold symmetry.

12. A = 2/51 = .0392+

Found by David Cantrell in June 2007.

3-fold dihedral symmetry.

13. A = .0306+

Found by David Cantrell in June 2007.

2-fold symmetry.

14. A = .0277+

Found by David Cantrell in June 2007.

2-fold symmetry.

15. A = .02456+

Found by Tej Stead in July 2026.

No symmetry.

16. A = .0222+

Found by David Cantrell in June 2007.

4-fold dihedral symmetry.

17. A = .018731+

Found by Tej Stead in July 2026.

No symmetry.

18. A = [2 – (3/2 – √(239/3)/6)1/3 – (3/2 + √(239/3)/6)1/3] / 18

A = .018238+

Found by Tej Stead in July 2026.

6-fold dihedral symmetry.

19. A = .014527+

Found by Tej Stead in July 2026.

No symmetry.

20. A = .013266+

Found by Tej Stead in July 2026.

No symmetry.