Erdős–Nagy theorem

The Erdős–Nagy theorem is a result in discrete geometry stating that a non-convex simple polygon can be made into a convex polygon by a finite sequence of flips. The flips are defined by taking a convex hull of a polygon and reflecting a pocket with respect to the boundary edge. The theorem is named after mathematicians Paul Erdős and Béla Szőkefalvi-Nagy.

History

Paul Erdős conjectured the result in 1935 as a problem in the American Mathematical Monthly, and Szőkefalvi-Nagy published a proof in 1939. The problem has a curious history and had been repeatedly rediscovered, until Branko Grünbaum surveyed the results in 1995. As it turns out, the original proof had a delicate mistake, which has been since corrected.

References

• Branko Grünbaum, How to convexify a polygon, Geombinatorics, 5 (1995), 24–30.
• Godfried Toussaint, The Erdős–Nagy Theorem and its Ramifications, Proc. 11th Canadian Conference on Computational Geometry (1999), 219–236.
• Branko Grünbaum and Joseph Zaks, Convexification of polygons by flips and by flipturns, Discrete Math. 241 (2001), 333–342.
• E.D. Demaine, B. Gassend, J. O'Rourke, G.T. Toussaint, All polygons flip finitely right? Surveys on discrete and computational geometry, 231–255, in Contemp. Math., 453, Amer. Math. Soc., Providence, RI, 2008.

External links

• The convexification of a simple polygon