On convexification of polygons by pops

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}.