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An n-Vertex Theorem for Convex Space Curves
Published online by Cambridge University Press: 20 November 2018
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The classical four-vertex theorem states that a simple closed convex C2 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 C3 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 P3.
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- Copyright © Canadian Mathematical Society 1985
References
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Scherk, P., Über Differenzierhare Kurven und Bögen I. Zum Begriff der Charakteristik, Časopis pěst. mat. a fys.
66 (1937), 165–171.Google Scholar
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