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

MathSciNet review:
760945

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Abstract: Let be a smooth closed plane curve given in general position. A bitangent of is, by definition, a line which is tangent to at two different points. Let and denote the numbers of all bitangents and all double points of , respectively. We prove here that if has no inflection points, . This is the affirmative answer to Halpern's conjecture.

**[1]**Thomas F. Banchoff,*Double tangency theorems for pairs of submanifolds*, Geometry Symposium, Utrecht 1980 (Utrecht, 1980) Lecture Notes in Math., vol. 894, Springer, Berlin-New York, 1981, pp. 26–48. MR**655418****[2]**Fr. Fabricius-Bjerre,*On the double tangents of plane closed curves*, Math. Scand**11**(1962), 113–116. MR**0161231****[3]**Benjamin Halpern,*Global theorems for closed plane curves*, Bull. Amer. Math. Soc.**76**(1970), 96–100. MR**0262936**, 10.1090/S0002-9904-1970-12380-1**[4]**Benjamin Halpern,*An inequality for double tangents*, Proc. Amer. Math. Soc.**76**(1979), no. 1, 133–139. MR**534404**, 10.1090/S0002-9939-1979-0534404-2**[5]**Hassler Whitney,*On regular closed curves in the plane*, Compositio Math.**4**(1937), 276–284. MR**1556973**

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DOI:
http://dx.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