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Torsion of locally convex curves


Author: Mohammad Ghomi
Journal: Proc. Amer. Math. Soc. 147 (2019), 1699-1707
MSC (2010): Primary 53A04, 53A05; Secondary 52A15, 53C45
DOI: https://doi.org/10.1090/proc/14367
Published electronically: January 9, 2019
MathSciNet review: 3910434
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Abstract: 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: ghomi@math.gatech.edu

Keywords: Torsion, vertex, inflection, osculating plane, tennis ball theorem.
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
Article copyright: © Copyright 2019 American Mathematical Society