Fenchel's theorem

In differential geometry, Fenchel's theorem (Werner Fenchel, 1929) states that the average curvature of any closed convex plane curve is

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

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.

where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is ${\displaystyle \geq {\frac {2\pi }{P}}}$ 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.