Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] T. Banchoff, Double tangency theorems for pairs of submanifolds, (Geometry Sympos., Utrecht, 1980), Lecture Notes in Math., vol. 894, Springer-Verlag, pp. 26-48. MR 655418 (83h:53005)
  • [2] Fr. Fabricius-Bjerre, On the double tangents of plane closed curves, Math. Scand. 11 (1962), 113-116. MR 0161231 (28:4439)
  • [3] B. Halpern, Global theorems for closed plane curves, Bull. Amer. Math. Soc. 76 (1970), 96-100. MR 0262936 (41:7541)
  • [4] -, An inequality for double tangents, Proc. Amer. Math. Soc. 76 (1979), 133-139. MR 534404 (80h:53002)
  • [5] H. Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276-284. MR 1556973

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A04, 52A10

Retrieve articles in all journals with MSC: 53A04, 52A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0760945-0
Keywords: Closed plane curve, bitangent, double point, inflection point, tangential degree
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society