Fenchel's theorem

A circle of 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 (Werner Fenchel, 1929) states that the average curvature of any closed convex plane curve is

${\frac {2\pi }{P}},$

where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is $\geq {\frac {2\pi }{P}}$ with equality holding only for convex plane curves.

References

• W. Fenchel (1929) Über Krümmung und Windung geschlossener Raumkurven, Mathematische Annalen 101: 238-252.
• Werner Fenchel (1951) The Differential Geometry of Closed Space Curves, Bulletin of the American Mathematical Society 57: 44 to 54, especially page 49, equation 13, via Project Euclid.