A lower bound for Heilbronn's triangle-problem

TL;DR

This paper constructs a set of n points on a unit circle demonstrating that the smallest triangle area formed by these points cannot be smaller than a bound of order n^(-3/2) times a logarithmic factor, providing a lower bound for Heilbronn's triangle problem.

Contribution

The paper presents a new lower bound construction for the minimal triangle area in Heilbronn's triangle problem on the circle.

Findings

Constructed point set on the circle with minimal triangle area of order n^(-3/2) (log n)^(-7/2)

Established a lower bound for the smallest triangle area in Heilbronn's problem

Improved understanding of the minimal triangle area bounds in convex domains

Abstract

Let n points be placed on a closed convex domain on the plane, no three points on a straight line. A conjecture by H. A. Heilbronn (before 1950) stated that on the convex domain of unit area the smallest triangle defined by these points has an area not larger than O(n^-2). Here is shown a construction of a set of n points on a unit circle where any of the triangles have an area not less than O(n^-3/2 * (log n)^-7/2).