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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Torsion of locally convex curves
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by Mohammad Ghomi PDF
Proc. Amer. Math. Soc. 147 (2019), 1699-1707 Request permission


We show that the torsion of any simple closed curve $\Gamma$ in Euclidean 3-space changes sign at least $4$ times provided that it is star-shaped and locally convex with respect to a point $o$ in the interior of its convex hull. The latter condition means that through each point $p$ of $\Gamma$ there passes a plane $H$, not containing $o$, such that a neighborhood of $p$ in $\Gamma$ lies on the same side of $H$ as does $o$. This generalizes the four vertex theorem of Sedykh for convex space curves. Following Thorbergsson and Umehara, we reduce the proof to the result of Segre on inflections of spherical curves, which is also known as Arnold’s tennis ball theorem.
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Additional Information
  • Mohammad Ghomi
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 687341
  • Email:
  • Received by editor(s): March 31, 2017
  • Received by editor(s) in revised form: May 31, 2018, and September 2, 2018
  • Published electronically: January 9, 2019
  • Additional Notes: Research of the author was supported in part by NSF grants DMS–1308777 and DMS -1711400.
  • Communicated by: Michael Wolf
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 1699-1707
  • MSC (2010): Primary 53A04, 53A05; Secondary 52A15, 53C45
  • DOI:
  • MathSciNet review: 3910434