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Thorbergsson, Gudlaugur and Umehara, Masaaki (2012). A refinement of Foreman's four-vertex theorem and its dual version. Kyoto J. Math., 52 (4). S. 743 - 759. DURHAM: DUKE UNIV PRESS. ISSN 2156-2261

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Abstract

A strictly convex curve is a C-infinity-regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve Gamma, and take two distinct tangent lines l(1) and l(2) of Gamma. A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of Gamma, which are tangent to l(1) and l(2), as a corollary of Ghys's theorem on diffeomorphisms of S-1. In this paper, we prove a refinement of Foreman's result. We then prove a projectively dual version of our refinement, which is a claim about semiosculating conics passing through two fixed points on Gamma. We also show that the dual version of Foreman's four-vertex theorem is almost equivalent to the Ghys's theorem.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Thorbergsson, Gudlaugur UNSPECIFIED UNSPECIFIED UNSPECIFIED Umehara, Masaaki UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-477079
DOI:	10.1215/21562261-1728848
Journal or Publication Title:	Kyoto J. Math.
Volume:	52
Number:	4
Page Range:	S. 743 - 759
Date:	2012
Publisher:	DUKE UNIV PRESS
Place of Publication:	DURHAM
ISSN:	2156-2261
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language SEXTACTIC POINTS; CONVEX OVAL; NUMBER Multiple languages Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/47707