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

2 π

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.

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.

Given a closed smooth curve

with unit speed, the velocity

is also a closed smooth curve (called tangent indicatrix).

The total absolute curvature is its length

does not lie in an open hemisphere.

lies in a closed hemisphere, then

is a plane curve.

By rotating the sphere, we may assume

are symmetric about the axis through the poles.

By the previous paragraph, at least one of the two curves

intersects with the equator at some point

We denote this curve by

, and the north pole, forming a closed curve

containing antipodal points

A curve connecting

, which is the length of the great semicircle between

, and if equality holds then

does not cross the equator.

, and if equality holds then

lies in a closed hemisphere, and thus

is a plane curve.