On the unfolding of simple closed curves

Pardon, John

doi:10.1090/S0002-9947-08-04781-8

On the unfolding of simple closed curves
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by John Pardon PDF

Trans. Amer. Math. Soc. 361 (2009), 1749-1764 Request permission

Abstract:

We show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons. We also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, we generalize two of the primary tools used in their program: the Farkas Lemma of linear programming to Banach spaces and the Maxwell-Cremona Theorem of rigidity theory to apply to stresses represented by measures on the plane.

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