On the unfolding of simple closed curves

Abstract

First let me quote from the official website. ``The Intel Science Talent Search (Intel STS), a program of the Society for Science \& the Public, is America's oldest and most prestigious pre-college science competition. The event brings together 40 of the best and brightest young scientific minds in America to compete for USD 1.25 million in awards and scholarships. Every year, roughly 1,600 U.S. high school seniors enter the Intel Science Talent Search with original science projects spanning a wide range of mathematics and science disciplines.'' In the 2007 Intel STS, the work in the article under review earned second place. ``For his mathematics project that solved a classical open problem in differential geometry, John Pardon of Chapel Hill, North Carolina's Durham Academy received a \$75,000 scholarship. John used a new approach to extend findings already known in polygons to a broad array of shapes. In his research, John was able to show that a finite-length closed curve in the plane can be made convex in a continuous manner, and without bringing any two points of the curve closer together.'' \par Specifically, this work extends [R. Connelly, E. D. Demaine\ and G. Rote, Discrete Comput. Geom. {\bf 30} (2003), no.~2, 205--239; [msn] MR2007962 (2004h:52028) [/msn]]. This extension is achieved by generalizing the Farkas Lemma of linear programming to Banach spaces, as well as the Maxwell-Cremona Theorem of rigidity theory to stresses represented by measures on the plane. In the end, it is shown that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion that preserves arc length without decreasing the ambient Euclidean distance between any pair of points on the curve.