# 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 Em geometria diferencial, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals {displaystyle 2pi /L} , Onde {estilo de exibição L} is the length of the curve. It is named after Werner Fenchel, que publicou em 1929. De forma geral, 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, Nova Iorque: Publicações de Dover, Inc. ISBN 978-0-486-80699-0. MR 3837152. Zbl 1352.53002. Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Anais Matemáticos (em alemão). 101 (1): 238-252. doi:10.1007/bf01454836. JFM 55.0394.06. MR 1512528. Fenchel, Werner (1951). "On the differential geometry of closed space curves". Boletim da American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. MR 0040040. Zbl 0042.40006.; see especially equation 13, página 49 O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdã: Imprensa Acadêmica. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. MR 2351345. Zbl 1208.53003. Spivak, Michael (1999uma). Uma introdução abrangente à geometria diferencial. Volume. III (Terceira edição do 1975 original ed.). Wilmington, DO: Publique ou pereça, Inc. ISBN 0-914098-72-1. MR 0532832. Zbl 1213.53001. Categorias: Theorems in differential geometryTheorems in plane geometryTheorems about curvesCurvature (matemática)

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