Fenchel's 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 ${\displaystyle 2\pi }$. Equivalently, the average curvature is at least ${\displaystyle 2\pi /L}$, where ${\displaystyle L}$ is the length of the curve. The only curves of this type whose total absolute curvature equals ${\displaystyle 2\pi }$ and whose average curvature equals ${\displaystyle 2\pi /L}$ are the plane convex curves. The theorem is named after Werner Fenchel, who published it in 1929.

Fenchel's theorem

Type	Theorem
Field	Differential geometry
Statement	A smooth closed space curve has total absolute curvature ${\displaystyle \geq 2\pi }$, with equality if and only if it is a convex plane curve
First stated by	Werner Fenchel
First proof in	1929

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.