# 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 In der Differentialgeometrie, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals {displaystyle 2pi /L} , wo {Anzeigestil L} is the length of the curve. It is named after Werner Fenchel, wer es veröffentlicht hat 1929. Allgemeiner, 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, Manfred P. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, NY: Dover-Veröffentlichungen, Inc. ISBN 978-0-486-80699-0. HERR 3837152. Zbl 1352.53002. Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Mathematische Annalen (auf Deutsch). 101 (1): 238–252. doi:10.1007/bf01454836. JFM 55.0394.06. HERR 1512528. Fenchel, Werner (1951). "On the differential geometry of closed space curves". Bulletin der American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. HERR 0040040. Zbl 0042.40006.; see especially equation 13, Seite 49 O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdam: Akademische Presse. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. HERR 2351345. Zbl 1208.53003. Spivak, Michael (1999a). Eine umfassende Einführung in die Differentialgeometrie. Vol. III (Dritte Auflage von 1975 original ed.). Wilmington, VON: Veröffentliche oder untergehe, Inc. ISBN 0-914098-72-1. HERR 0532832. Zbl 1213.53001. Kategorien: Theorems in differential geometryTheorems in plane geometryTheorems about curvesCurvature (Mathematik)

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