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Proceedings of the American Mathematical Society

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

 
 

 

On Halpern’s conjecture for closed plane curves


Author: Tetsuya Ozawa
Journal: Proc. Amer. Math. Soc. 92 (1984), 554-560
MSC: Primary 53A04; Secondary 52A10
DOI: https://doi.org/10.1090/S0002-9939-1984-0760945-0
MathSciNet review: 760945
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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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Keywords: Closed plane curve, bitangent, double point, inflection point, tangential degree
Article copyright: © Copyright 1984 American Mathematical Society