Four part theorem

Four vertex theorem for the ellipse

The four-vertex theorem is a set of differential geometry about curves in the plane . It says that for every simply closed, smooth, flat curve, the curvature function has at least four extreme points . Points of a curve at which the curvature has a local extreme (i.e. a local maximum or minimum ) are called vertices (see vertex ).

The theorem was proved for convex curves in 1909 by the Indian mathematician Syamadas Mukhopadhyaya (1866–1937) and in the general case by Adolf Kneser in 1912.

There is also an inverse theorem: every continuous real function on the circle with at least two local maxima and two local minima is the curvature function of a plane simple closed curve. The theorem was proved for positive definite functions in 1971 by Herman Gluck and in the general case by Björn Dahlberg (1998).

literature

Christian Bär : Elementary Differential Geometry. Walter de Gruyter, Berlin et al. 2001, ISBN 3-11-015519-2 , p. 57 ( excerpt (Google) ).
Dennis DeTurck, Herman Gluck, Daniel Pomerleano, David Shea Vick: The Four Vertex Theorem and Its Converse (PDF file; 1.47 MB). In: Notices of the American Mathematical Society. Vol. 54, No. February 2, 2007, ISSN 0002-9920 , pp. 192-207.
WC Graustein: Extensions of the Four-Vertex Theorem . Transactions of the American Mathematical Society, Vol. 41, No. 1 (Jan. 1937), pp. 9-23 ( JSTOR 1989876 )

Web links

Britta Meixner, Ana-Catalina Plesa: Differential Geometry (PDF file; 1.91 MB). Lecture notes at the University of Passau, July 2006, pp. 31–46
Sebastian Klein: curves and surfaces . Lecture notes at the University of Mannheim, winter semester 2008, p. 35

Individual evidence

↑ Mukhopadhyaya: New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc. 1, 1909, 21-27
↑ ^a ^b ^c ^d Dennis DeTurck, Herman Gluck, Daniel Pomerleano, David Shea Vick: The Four Vertex Theorem and Its Converse (PDF file; 1.47 MB). In: Notices of the American Mathematical Society. Vol. 54, No. February 2, 2007, ISSN 0002-9920 , pp. 192-207.
↑ Kneser: Remarks on the number of extremes of curvature on closed curves and on related questions in a non-Euclidean geometry, Festschrift Heinrich Weber. Teubner. 1912, pp. 170-180
↑ Gluck, The converse to the four-vertex theorem, L'Enseignement Math. 17, 1971, 295-309
↑ Dahlberg, The converse of the four vertex theorem, Proc. Amer. Math. Soc. 133, 2005, pp. 2131-2135

[1] Mukhopadhyaya: New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc. 1, 1909, 21-27

[AMS-2] Dennis DeTurck, Herman Gluck, Daniel Pomerleano, David Shea Vick: The Four Vertex Theorem and Its Converse (PDF file; 1.47 MB). In: Notices of the American Mathematical Society. Vol. 54, No. February 2, 2007, ISSN 0002-9920 , pp. 192-207.

[3] Kneser: Remarks on the number of extremes of curvature on closed curves and on related questions in a non-Euclidean geometry, Festschrift Heinrich Weber. Teubner. 1912, pp. 170-180

[4] Gluck, The converse to the four-vertex theorem, L'Enseignement Math. 17, 1971, 295-309

[5] Dahlberg, The converse of the four vertex theorem, Proc. Amer. Math. Soc. 133, 2005, pp. 2131-2135