Fenchel's theorem

A circle with radius r has average curvature 1/r=2π/P, where P=2πr is the perimeter.

This article is about the concept in geometry. For the concept in mathematical optimization, see Fenchel's duality theorem.

In differential geometry, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals $2\pi /L$, where $L$ is the length of the curve. It is named after Werner Fenchel, who published it in 1929. More generally, for an arbitrary closed space curve the average curvature is $\geq 2\pi /L$ 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, NY: Dover Publications, Inc. ISBN 978-0-486-80699-0. MR 3837152. Zbl 1352.53002.
• Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Mathematische Annalen (in German). 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". Bulletin of the American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. MR 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: Academic Press. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. MR 2351345. Zbl 1208.53003.
• Spivak, Michael (1999a). A comprehensive introduction to differential geometry. Vol. III (Third edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc. ISBN 0-914098-72-1. MR 0532832. Zbl 1213.53001.