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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the unfolding of simple closed curves


Author: John Pardon
Journal: Trans. Amer. Math. Soc. 361 (2009), 1749-1764
MSC (2000): Primary 53C24; Secondary 53A04
Published electronically: November 5, 2008
MathSciNet review: 2465815
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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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Additional Information

John Pardon
Affiliation: Durham Academy Upper School, 3601 Ridge Road, Durham, North Carolina 27705
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: jpardon@princeton.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04781-8
PII: S 0002-9947(08)04781-8
Received by editor(s): December 29, 2006
Published electronically: November 5, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.