Fenchel's theorem

This article is about the concept in geometry. For the concept in mathematical optimization, see Fenchel's duality theorem.

Fenchel's theorem

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

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 $2\pi$. Equivalently, the average curvature is at least $2\pi /L$, where $L$ is the length of the curve. The only curves of this type whose total absolute curvature equals $2\pi$ and whose average curvature equals $2\pi /L$ 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.

Proof

Given a closed smooth curve $\alpha :[0,L]\to \mathbb {R} ^{3}$ with unit speed, the velocity $\gamma ={\dot {\alpha }}:[0,L]\to \mathbb {S} ^{2}$ is also a closed smooth curve. The total absolute curvature is its length $l(\gamma )$.

The curve $\gamma$ does not lie in an open hemisphere. If so, then there is $v\in \mathbb {S} ^{2}$ such that $\gamma \cdot v>0$, so $\textstyle 0=(\alpha (1)-\alpha (0))\cdot v=\int _{0}^{L}\gamma (t)\cdot v\,\mathrm {d} t>0$, a contradiction. This also shows that if $\gamma$ lies in a closed hemisphere, then $\gamma \cdot v\equiv 0$, so $\alpha$ is a plane curve.

Consider a point $\gamma (T)$ such that curves $\gamma ([0,T])$ and $\gamma ([T,L])$ have the same length. By rotating the sphere, we may assume $\gamma (0)$ and $\gamma (T)$ are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves $\gamma ([0,T])$ and $\gamma ([T,L])$ intersects with the equator at some point $p$. We denote this curve by $\gamma _{0}$. Then $l(\gamma )=2l(\gamma _{0})$.

We reflect $\gamma _{0}$ across the plane through $\gamma (0)$, $\gamma (T)$, and the north pole, forming a closed curve $\gamma _{1}$ containing antipodal points $\pm p$, with length $l(\gamma _{1})=2l(\gamma _{0})$. A curve connecting $\pm p$ has length at least $\pi$, which is the length of the great semicircle between $\pm p$. So $l(\gamma _{1})\geq 2\pi$, and if equality holds then $\gamma _{0}$ does not cross the equator.

Therefore, $l(\gamma )=2l(\gamma _{0})=l(\gamma _{1})\geq 2\pi$, and if equality holds then $\gamma$ lies in a closed hemisphere, and thus $\alpha$ is a plane curve.

References

• do Carmo, Manfredo P. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, NY: Dover Publications, Inc. ISBN 978-0-486-80699-0. MR 3837152. Zbl 1352.53002.
• Fenchel, Werner (1929). "Über Krümmung und Windung geschlossener Raumkurven". Mathematische Annalen (in German). 101 (1): 238–252. doi:10.1007/bf01454836. JFM 55.0394.06. MR 1512528. S2CID 119908321.
• Fenchel, Werner (1951). "On the differential geometry of closed space curves". Bulletin of the American Mathematical Society. 57 (1): 44–54. doi:10.1090/S0002-9904-1951-09440-9. MR 0040040. Zbl 0042.40006.; see especially equation 13, page 49
• O'Neill, Barrett (2006). Elementary differential geometry (Revised second edition of 1966 original ed.). Amsterdam: Academic Press. doi:10.1016/C2009-0-05241-6. ISBN 978-0-12-088735-4. MR 2351345. Zbl 1208.53003.
• Spivak, Michael (1999). A comprehensive introduction to differential geometry. Vol. III (Third edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc. ISBN 0-914098-72-1. MR 0532832. Zbl 1213.53001.