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

TL;DR

This work tackles the Heilbronn-type problem for $k$-gons by introducing a rectangle-based reduction and a recursive, pigeonhole-driven framework to bound the minimal $k$-gon area. The authors prove the sharp asymptotic upper bound $\Delta_4(n) \le \dfrac{2}{n} + o\left(\dfrac{1}{n}\right)$ and generalize to $\Delta_k(n) \le \dfrac{k-2}{n} + o\left(\dfrac{1}{n}\right)$ for all fixed $k\ge4$, extending the classic trivial bound $3/n$. Central to the approach is the auxiliary function $\Delta'_k(n)$ on unit rectangles, together with a recursive partitioning argument that preserves upper bounds while moving from $n$ to smaller subinstances. The results advance the understanding of discrete discrepancy-type problems and hold for arbitrary convex figures of unit area, with conjectures about tightness guided by grid-based lower bounds.

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 $Ω(n^{-3/2})$, which was improved to $Ω(n^{-3/2}\log{n})$ 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{é}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.

Figures (2)

• Figure 3.1: $\Delta_4'(6)\geq 1/2$
• Figure 3.2: $\Delta_4'(9)\geq 1/4$

Theorems & Definitions (11)

• Theorem 2.1
• Theorem 2.2
• Corollary 3.3
• Lemma 3.5
• Theorem 3.6
• Corollary 3.8
• Conjecture 3.9
• Proposition 4.1: Analogue of Observation \ref{['phpobs']}
• Theorem 4.2: Analogue of Theorem \ref{['recurse_theorem']}
• Corollary 4.4