On Halpern’s conjecture for closed plane curves

Ozawa, Tetsuya

doi:10.1090/S0002-9939-1984-0760945-0

On Halpern’s conjecture for closed plane curves
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by Tetsuya Ozawa

Proc. Amer. Math. Soc. 92 (1984), 554-560

DOI: https://doi.org/10.1090/S0002-9939-1984-0760945-0

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Abstract:

Let $c$ be a smooth closed plane curve given in general position. A bitangent of $c$ is, by definition, a line which is tangent to $c$ at two different points. Let $B(c)$ and $D(c)$ denote the numbers of all bitangents and all double points of $c$, respectively. We prove here that if $c$ has no inflection points, $B(c) \leqslant D(c)(2D(c) - 1)$. This is the affirmative answer to Halpern’s conjecture.

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