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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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.
References
  • Jason H. Cantarella, Erik D. Demaine, Hayley N. Iben, and James F. O’Brien, An energy-driven approach to linkage unfolding, SCG ’04: Proceedings of the twentieth annual symposium on Computational Geometry (New York, NY, USA), ACM Press, 2004, pp. 134–143.
  • Robert Connelly, Erik D. Demaine, and Günter Rote, Straightening polygonal arcs and convexifying polygonal cycles, Discrete Comput. Geom. 30 (2003), no. 2, 205–239. U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000). MR 2007962, DOI 10.1007/s00454-003-0006-7
  • Mohammad Ghomi, Classical open problems in differential geometry, 2004.
  • O. Hernandez-Lerma and J. B. Lasserre, Cone-constrained linear equations in Banach spaces, J. Convex Anal. 4 (1997), no. 1, 149–164. MR 1459886
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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
  • MR Author ID: 857067
  • Email: jpardon@princeton.edu
  • Received by editor(s): December 29, 2006
  • Published electronically: November 5, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 1749-1764
  • MSC (2000): Primary 53C24; Secondary 53A04
  • DOI: https://doi.org/10.1090/S0002-9947-08-04781-8
  • MathSciNet review: 2465815