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.

翻译：Heilbronn 三角形问题要求在单位正方形内放置 $n$ 个点，最大化任意三点所构成的三角形的最小面积。1972 年，Schmidt 考虑该问题的自然推广：在单位正方形内放置 $n$ 个点，最大化任意四点所构成的凸包的最小面积。他给出了 $\Omega(n^{-3/2})$ 的下界，随后 Leffman 将其改进为 $\Omega(n^{-3/2}\log{n})$。平凡上界可达 $3/n$，Schmidt 询问是否能对该上界进行渐近改进。然而，尽管经过多次尝试，过去 50 年间该平凡上界未有任何渐近改进，此问题逐渐被视为公认的难题。Szemerédi 提出能否至少改进该平凡上界中的常数。本研究通过证明上界 $2/n+o(1/n)$ 回答了该问题，并将结论推广至任意 $k\geq 4$ 点构成的凸包情形。