On the Existence of $U$-Polygons of Class $c\geq 4$ in Planar Point Sets

TL;DR

This paper characterizes the existence and properties of $U$-polygons of class $c \,\geq\, 4$ in specific planar point sets, including explicit results for cyclotomic model sets, advancing understanding of geometric configurations related to direction sets.

Contribution

It provides a characterization of the edge counts of $U$-polygons of class $c\geq 4$ and derives explicit results for cyclotomic model sets, extending previous geometric and combinatorial knowledge.

Findings

Characterization of $U$-polygons of class $c\geq 4$ in certain subsets of the plane.

Explicit results for $U$-polygons in cyclotomic model sets.

Conditions under which $U$-polygons of specified class exist.

Abstract

For a finite set $U$ of directions in the Euclidean plane, a convex non-degenerate polygon $P$ is called a $U$-polygon if every line parallel to a direction of $U$ that meets a vertex of $P$ also meets another vertex of $P$. We characterize the numbers of edges of $U$-polygons of class $c\geq4$ with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets.