# 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 geometria differenziale, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals {displaystyle 2pi /L} , dove {stile di visualizzazione L} is the length of the curve. It is named after Werner Fenchel, in chi l'ha pubblicato 1929. Più generalmente, 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: Pubblicazioni di Dover, Inc. ISBN 978-0-486-80699-0. SIG 3837152. Zbl 1352.53002. finocchio, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Annali matematici (in tedesco). 101 (1): 238–252. doi:10.1007/bf01454836. JFM 55.0394.06. SIG 1512528. finocchio, Werner (1951). "On the differential geometry of closed space curves". Bollettino dell'American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. SIG 0040040. Zbl 0042.40006.; see especially equation 13, pagina 49 O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdam: Stampa accademica. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. SIG 2351345. Zbl 1208.53003. Spivak, Michael (1999un). Un'introduzione completa alla geometria differenziale. vol. III (Terza edizione di 1975 original ed.). Wilmington, DI: Pubblica o muori, Inc. ISBN 0-914098-72-1. SIG 0532832. Zbl 1213.53001. Categorie: Theorems in differential geometryTheorems in plane geometryTheorems about curvesCurvature (matematica)

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