Fenchel's theorem

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

In differential geometry, Fenchel's theorem states that the average curvature of any closed convex curve in the Euclidean plane equals ${\displaystyle 2\pi /L}$, where ${\displaystyle L}$ 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 ${\displaystyle \geq 2\pi /L}$ with equality holding only for convex plane curves.

References[]

• Fenchel, Werner (December 1929), "Über Krümmung und Windung geschlossener Raumkurven", Mathematische Annalen (in German), 101 (1): 238–252, doi:10.1007/bf01454836
• Fenchel, Werner (1951), "On the differential geometry of closed space curves", Bulletin of the American Mathematical Society, 57: 44–54, doi:10.1090/S0002-9904-1951-09440-9, MR 0040040; see especially equation 13, page 49