## Duke Mathematical Journal

### Tangent lines, inflections, and vertices of closed curves

#### Abstract

We show that every smooth closed curve $\Gamma$ immersed in Euclidean space $\mathbf {R}^{3}$ satisfies the sharp inequality $2(\mathcal{P}+\mathcal{I})+\mathcal{V}\geq6$ which relates the numbers $\mathcal{P}$ of pairs of parallel tangent lines, $\mathcal{I}$ of inflections (or points of vanishing curvature), and $\mathcal{V}$ of vertices (or points of vanishing torsion) of $\Gamma$. We also show that $2(\mathcal{P^{+}}+\mathcal{I})+\mathcal{V}\geq4$, where $\mathcal{P}^{+}$ is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve-shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane $\mathbf {RP}^{2}$ and the sphere $\mathbf {S}^{2}$ which intersect every closed geodesic. These findings extend some classical results in curve theory from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”

#### Article information

Source
Duke Math. J., Volume 162, Number 14 (2013), 2691-2730.

Dates
Revised: 16 February 2013
First available in Project Euclid: 6 November 2013

https://projecteuclid.org/euclid.dmj/1383760702

Digital Object Identifier
doi:10.1215/00127094-2381038

Mathematical Reviews number (MathSciNet)
MR3127811

Zentralblatt MATH identifier
1295.53002

#### Citation

Ghomi, Mohammad. Tangent lines, inflections, and vertices of closed curves. Duke Math. J. 162 (2013), no. 14, 2691--2730. doi:10.1215/00127094-2381038. https://projecteuclid.org/euclid.dmj/1383760702

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