On convexification of polygons by pops

TL;DR

This paper investigates whether polygons can be made convex through a series of vertex reflections called pops, and proves that certain families of polygons, including self-intersecting ones, cannot be convexified by these operations.

Contribution

The paper introduces a family of polygons and proves they cannot be convexified by pop operations, answering an open problem in computational geometry.

Findings

Certain polygons cannot be convexified by pops

The family includes simple and self-intersecting polygons

Answers an open problem by Demaine and O'Rourke

Abstract

Given a polygon $P$ in the plane, a {\em pop} operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an open problem posed by Demaine and O'Rourke \cite[Open Problem 5.3]{DO07}.