Fenchel's theorem
Gives the average curvature of any closed convex plane curve / From Wikipedia, the free encyclopedia
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 is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least . Equivalently, the average curvature is at least , where is the length of the curve. The only curves of this type whose total absolute curvature equals and whose average curvature equals are the plane convex curves. The theorem is named after Werner Fenchel, who published it in 1929.
Quick Facts Type, Field ...
Type | Theorem |
---|---|
Field | Differential geometry |
Statement | A smooth closed space curve has total absolute curvature , with equality if and only if it is a convex plane curve |
First stated by | Werner Fenchel |
First proof in | 1929 |
Close
The Fenchel theorem is enhanced by the Fáry–Milnor theorem, which says that if a closed smooth simple space curve is nontrivially knotted, then the total absolute curvature is greater than 4π.