The Erdös-Nagy theorem and its ramifications.
Godfried T. Toussaint · 1999
Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erdős introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 B'ela Nagy pointed out that flipping several pockets simultaneously may result in a nonsimple polygon. Modifying the problem slightly he then proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. We call this result the Erdős-Nagy Theorem. Since then this theorem has been rediscovered many times in different contexts, apparently, with none of the authors aware of each other's work. One purpose of this paper is to bring to light this "hidden" work...