Fenchel's theorem

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

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

where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is with equality holding only for convex plane curves.


  • W. Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929), 238-252. [1]