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
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 Eric W. Weisstein