The classical four-vertex theorem states that a simple closed convex C 2 curve in the Euclidean plane has at least four vertices (points of extreme curvature). This theorem has many generalizations with regard to both the curve and the topological space and for a history of the subject, we refer to [4] and [1]. The particular generalization of concern, credited to H. Mohrmann, is the following n-vertex theorem.Let a simple closed C 3 curve on a closed convex surface be intersected by a suitable plane in n points. Then the curve has at least n inflections (vertices).The closed convex surface in the preceding is defined as having at most two points in common with any straight line. Presently, we extend this result to curves on more general convex surfaces in a real projective three-space P 3.
Bisztriczky, Tibor. An n-Vertex Theorem for Convex Space Curves. Canadian journal of mathematics, Tome 37 (1985) no. 2, pp. 217-237. doi: 10.4153/CJM-1985-014-7
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author = {Bisztriczky, Tibor},
title = {An {n-Vertex} {Theorem} for {Convex} {Space} {Curves}},
journal = {Canadian journal of mathematics},
pages = {217--237},
year = {1985},
volume = {37},
number = {2},
doi = {10.4153/CJM-1985-014-7},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1985-014-7/}
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AU - Bisztriczky, Tibor
TI - An n-Vertex Theorem for Convex Space Curves
JO - Canadian journal of mathematics
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