# Fenchel's theorem

Fenchel's theorem This article is about the concept in geometry. For the concept in mathematical optimization, see Fenchel's duality theorem. Fenchel's theorem Type Theorem Field Differential geometry Statement A circle with radius r has average curvature 1/r=2π/P, where P=2πr is the perimeter. Conjectured by Werner Fenchel Conjectured in 1929 En géométrie différentielle, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals {displaystyle 2pi /L} , où {displaystyle L} is the length of the curve. It is named after Werner Fenchel, qui l'a publié dans 1929. Plus généralement, for an arbitrary closed space curve the average curvature is {displaystyle geq 2pi /L} with equality holding only for convex plane curves.

References do Carmo, Manfredo P.. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, New York: Publications de Douvres, Inc. ISBN 978-0-486-80699-0. M 3837152. Zbl 1352.53002. Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Annales mathématiques (en allemand). 101 (1): 238–252. est ce que je:10.1007/bf01454836. JFM 55.0394.06. M 1512528. Fenchel, Werner (1951). "On the differential geometry of closed space curves". Bulletin de l'American Mathematical Society. 57 (1): 44–54. est ce que je:10.1090/S0002-9904-1951-09440-9. M 0040040. Zbl 0042.40006.; see especially equation 13, page 49 O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdam: Presse académique. est ce que je:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. M 2351345. Zbl 1208.53003. Spivak, Michael (1999un). Une introduction complète à la géométrie différentielle. Volume. III (Troisième édition de 1975 original ed.). Wilmington, DE: Publier ou périr, Inc. ISBN 0-914098-72-1. M 0532832. Zbl 1213.53001. Catégories: Theorems in differential geometryTheorems in plane geometryTheorems about curvesCurvature (mathématiques)

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