The flips are defined by taking a convex hull of a polygon and reflecting a pocket with respect to the boundary edge.
The Erdős–Nagy theorem states that it is always possible to find a sequence of flips that produces a convex polygon in this way.
There exist quadrilaterals that require an arbitrarily large (but finite) number of flips to be made convex.
Additional proofs (some but not all correct) were provided in 1957 by two independent Russian mathematicians, Reshetnyak and Yusupov, in 1959, by Bing and Kazarinoff, and in 1993 by Wegner.
Demaine, Gassend, O'Rourke, and Toussaint survey this history and provide a corrected proof.