Improved upper bounds for the Heilbronn's Problem for $k$-gons
<|reference_start|>Improved upper bounds for the Heilbronn's Problem for $k$-gons: 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.<|reference_end|>
@article{gajjala2024improved,
    title={Improved upper bounds for the Heilbronn's Problem for $k$-gons},
    author={Rishikesh Gajjala and Jayanth Ravi},
    journal={arXiv preprint arXiv:2405.12945},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.12945},
    primaryClass={cs.DM cs.CG math.CO}
}