Improved upper bounds for the Heilbronn's Problem for $k$-gons

TL;DR

This paper improves the asymptotic upper bound for the maximum minimal convex hull area of four points in a unit square, advancing understanding of a longstanding generalization of Heilbronn's problem.

Contribution

The authors establish a new upper bound of 2/n + o(1/n) for the minimal convex hull area, extending results to convex hulls of any size k ≥ 4.

Findings

Proved an upper bound of 2/n + o(1/n) for the problem.

Extended the upper bound results to convex hulls of size k ≥ 4.

Resolved a 50-year-old open question about the asymptotic behavior of the problem.

Abstract

The Heilbronn triangle problem asks for the placement of $n$ points in a unit square that maximizes the smallest area of a triangle formed by any three of those points. In $1972$, Schmidt considered a natural generalization of this problem. He asked for the placement of $n$ points in a unit square that maximizes the smallest area of the convex hull formed by any four of those points. He showed a lower bound of $\Omega(n^{-3/2})$, which was improved to $\Omega(n^{-3/2}\log{n})$ by Leffman. A trivial upper bound of $3/n$ could be obtained, and Schmidt asked if this could be improved asymptotically. However, despite several efforts, no asymptotic improvement over the trivial upper bound was known for the last $50$ years, and the problem started to get the tag of being notoriously hard. Szemer{\'e}di posed the question of whether one can, at least, improve the constant in this trivial upper bound. In this work, we answer this question by proving an upper bound of $2/n+o(1/n)$. We also extend our results to any convex hulls formed by $k\geq 4$ points.