Pentagon Minimization without Computation

Abstract

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.