Erd\H{o}s and Guy initiated a line of research studying $\mu_k(n)$, the minimum number of convex $k$-gons one can obtain by placing $n$ points in the plane without any three of them being collinear. Asymptotically, the limits $c_k := \lim_{n\to \infty} \mu_k(n)/\binom{n}{k}$ exist for all $k$, and are strictly positive due to the Erd\H{o}s-Szekeres theorem. This article focuses on the case $k=5$, where $c_5$ was known to be between $0.0608516$ and $0.0625$ (Goaoc et al., 2018; Subercaseaux et al., 2023). The lower bound was obtained through the Flag Algebra method of Razborov using semi-definite programming. In this article we prove a more modest lower bound of $\frac{5\sqrt{5}-11}{4} \approx 0.04508$ without any computation; we exploit``planar-point equations'' that count, in different ways, the number of convex pentagons (or other geometric objects) in a point placement. To derive our lower bound we combine such equations by viewing them from a statistical perspective, which we believe can be fruitful for other related problems.

翻译：Erdős 和 Guy 开创了一系列研究，探讨通过将 n 个点放置在平面上且任意三点不共线时，所能得到的最小凸 k 边形数量 μ_k(n)。渐近地，对所有 k，极限 c_k := lim_{n→∞} μ_k(n)/\binom{n}{k} 均存在，并且根据 Erdős-Szekeres 定理，这些极限严格为正。本文聚焦于 k=5 的情形，其中已知 c_5 介于 0.0608516 与 0.0625 之间（Goaoc 等人，2018；Subercaseaux 等人，2023）。下界是通过 Razborov 的旗代数方法结合半定规划获得的。在本文中，我们证明了一个更保守的下界：\(\frac{5\sqrt{5}-11}{4} \approx 0.04508\)，且无需任何计算；我们利用了“平面点方程”，这些方程以不同方式计算点布局中凸五边形（或其他几何对象）的数量。为了推导我们的下界，我们从统计视角出发组合这些方程，我们相信这种方法对于其他相关问题也可能富有成效。