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(n3/2) 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.
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BibTeX
@misc{nicolas2026visibility,
title = {{Visibility Sets, Empty Pentagons, and Sufficient Conditions for Hexagons}},
author = {Carlos M. Nicolas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23553},
eprint = {2610.23553},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23553},
}
RIS
TY - UNPB
TI - Visibility Sets, Empty Pentagons, and Sufficient Conditions for Hexagons
AU - Carlos M. Nicolas
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23553
UR - https://arxiv.org/abs/2610.23553
AB - 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.
ER -