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Torsion of locally convex curves
Differential Geometry · Submitted 31 Mar 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.98 | 0.95 | 0.96 | asserted 1.00 |
| s1 | 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 $o$. | background 0.74 | 0.11 | 0.03 | asserted 0.97 |
| s2 | This generalizes the four vertex theorem of Sedykh for convex space curves. | interpretation 0.90 | 0.52 | 0.04 | asserted 1.00 |
| s3 | 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. | method 0.99 | 0.78 | 0.02 | asserted 1.00 |
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The paper
Original 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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