Visibility Sets, Empty Pentagons, and Sufficient Conditions for Hexagons

Abstract

Given a set $S$ of $n$ points in general position in the plane, we define the $C$-set $C(p,q)$ of an oriented edge $(p,q)$ as the set of extreme points of the visibility set $V(p,q)$ facing $(p,q)$. Since every point of $V(p,q)$ that is not in $C(p,q)$ gives rise to an empty convex pentagon having $(p,q)$ as a boundary edge, it is natural to study the properties of $C$-sets in order to bound the number of empty convex pentagons of $S$. We prove tight bounds on the intersection size and union complexity of $C$-sets sharing an endpoint. We also consider the problem of bounding the number of empty pentagons and hexagons in terms of the number of interior and boundary points of $S$. While the hexagon bound is weak, it offers a potential step toward a superlinear lower bound. On the other hand, the two-parameter bound for pentagons, combined with the structural results on $C$-sets, yields a simple proof of an $Ω(n^{3/2})$ bound for the number of pentagons. While this is a significant improvement over previously published bounds, a recent preprint \cite{Astudillo2026} attains a better bound using completely different methods. Thus, the primary contribution of this paper lies in the structural results about $C$-sets and the two-parameter bounds for empty pentagons and hexagons.