Pentagon Minimization without Computation

TL;DR

This paper establishes a new, non-computational lower bound for the minimum number of convex pentagons in large point sets, using planar-point equations and a statistical approach, advancing understanding in combinatorial geometry.

Contribution

It introduces a novel, computation-free method to derive lower bounds for convex polygons in point sets, focusing on pentagons, through planar-point equations and statistical combination.

Findings

Proves a lower bound of approximately 0.04508 for convex pentagons without computation.

Uses planar-point equations and statistical methods to analyze geometric configurations.

Provides a new approach that could be applied to related combinatorial geometry problems.

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.