Heilbronn triangle problem

In mathematics, the Heilbronn triangle problem is a typical question in the area of irregularities of distribution, within elementary geometry.

Consider region "D" in the plane: a unit circle or general polygon — the asymptotics of the problem, which are the interesting aspect, aren't dependent on the exact shape. Place a number "n" of distinct points (greater than three) within "D": there are numerous triangles that can be constructed from three of those points. For a given configuration, we are interested knowing that there is some small triangle, and "how small". The Heilbronn triangle problem involves therefore the extremal case: minimum area from these points. The question was posed by Hans Heilbronn, of giving a lower bound for this minimum area, denoted by Δ("n"). This is therefore formally of the shape

:Δ("n") = max_{"X" in "C"("D","n")} min_{triangles "T" of "X"} Area of "T".

The notation here says that "X" is a configuration of "n" points in "D", and "T" is a triangle with three points of "X" as vertices.

Heilbronn initially conjectured that this area would be of order

:Δ("n") ~ "Constant"·"n"⁻²;

however there have been improvements to this bound.

As of 2004 it is known that

:"A"·"n"⁻²log "n" ≤ Δ("n") ≤ "B"·"n"^−8/7exp("C"√log "n")

where "A", "B" and "C" are constants.

References

*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.

External links

* [http://mathworld.wolfram.com/HeilbronnTriangleProblem.html Heilbronn Triangle Problem (MathWorld)]
* [http://www.mathsoft.com/mathresources/constants/geometryconstant/article/0,,2056,00.html Literature references]