*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Given any arrangement of points within a Unit Square, let be the smallest value for which
there is at least one Triangle formed from three of the points with Area . The first few
values are

Komlós

for any and all sufficiently large .

Using an Equilateral Triangle of unit Area instead gives the constants

**References**

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hlb/hlb.html

Goldberg, M. ``Maximizing the Smallest Triangle Made by Points in a Square.'' *Math. Mag.* **45**, 135-144, 1972.

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 242-244, 1994.

Komlos, J.; Pintz, J.; and Szemerédi, E. ``On Heilbronn's Triangle Problem.'' *J. London Math. Soc.*
**24**, 385-396, 1981.

Komlos, J.; Pintz, J.; and Szemerédi, E. ``A Lower Bound for Heilbronn's Triangle Problem.''
*J. London Math. Soc.* **25**, 13-24, 1982.

Roth, K. F. ``Developments in Heilbronn's Triangle Problem.'' *Adv. Math.* **22**, 364-385, 1976.

© 1996-9

1999-05-25