Heilbronn triangle problem
Encyclopedia
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
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
however there have been improvements to this bound.
it is known that
where A, B and C are constants.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
or general polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
— 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
Hans Heilbronn
Hans Arnold Heilbronn was a mathematician.He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate...
, of giving a lower bound for this minimum area, denoted by Δ(n). This is therefore formally of the shape
- Δ(n) = maxX in C(D,n) mintriangles 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.
it is known that
- A·n−2log n ≤ Δ(n) ≤ B·n−8/7exp(C√log n)
where A, B and C are constants.