Fenchel's theorem

{{short description|Gives the average curvature of any closed convex plane curve}}

{{About|the concept in geometry|the concept in mathematical optimization|Fenchel's duality theorem}}

{{Infobox mathematical statement

| name = Fenchel's theorem

| image =

| statement = A smooth closed space curve has total absolute curvature $\ge 2\pi$ , with equality if and only if it is a convex plane curve

| first stated by = Werner Fenchel

| first proof date = 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 {{math|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 (called tangent indicatrix). 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 $\textstyle0=(\alpha(1)-\alpha(0))\cdot v=\int_0^L\gamma(t)\cdot v\,\mathrm{d}t>0$ , a contradiction. This also shows that if $\gamma$ lies in a closed hemisphere, then $\gamma\cdot v\equiv0$ , 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 $l(\gamma_1)\ge2\pi$ , and if equality holds then $\gamma_0$ does not cross the equator.

Therefore, $l(\gamma)=2l(\gamma_0)=l(\gamma_1)\ge2\pi$ , and if equality holds then $\gamma$ lies in a closed hemisphere, and thus $\alpha$ is a plane curve.

References

{{cite book|mr=3837152|last1=do Carmo|first1=Manfredo P.|author-link1=Manfredo do Carmo|title=Differential geometry of curves & surfaces|edition=Revised & updated second edition of 1976 original|publisher=Dover Publications, Inc.|location=Mineola, NY|year=2016|isbn=978-0-486-80699-0|zbl=1352.53002}}
{{cite journal

| last = Fenchel | first = Werner | author-link = Werner Fenchel

| date = 1929

| doi = 10.1007/bf01454836

| issue = 1

| journal = Mathematische Annalen

| language = de

| pages = 238–252

| title = Über Krümmung und Windung geschlossener Raumkurven

| volume = 101|mr=1512528|jfm=55.0394.06| s2cid = 119908321 |url=https://eudml.org/doc/159330}}

{{cite journal

| last = Fenchel | first = Werner | author-link = Werner Fenchel

| doi = 10.1090/S0002-9904-1951-09440-9

| journal = Bulletin of the American Mathematical Society

| mr = 0040040

| pages = 44–54

| title = On the differential geometry of closed space curves

| volume = 57

| year = 1951| doi-access = free|zbl=0042.40006|issue=1

}}; see especially equation 13, page 49

{{cite book|title=Elementary differential geometry|first=Barrett|last=O'Neill|author-link1=Barrett O'Neill|edition=Revised second edition of 1966 original|location=Amsterdam|publisher=Academic Press|mr=2351345|year=2006|isbn=978-0-12-088735-4|zbl=1208.53003|doi=10.1016/C2009-0-05241-6}}
{{cite book|mr=0532832|last1=Spivak|first1=Michael|title=A comprehensive introduction to differential geometry. Vol. III|edition=Third edition of 1975 original|publisher=Publish or Perish, Inc.|location=Wilmington, DE|year=1999|isbn=0-914098-72-1|author-link1=Michael Spivak|zbl=1213.53001}}
{{cite web | url = https://www.math.brown.edu/tbanchof/stanton/stanton.html | author = Thomas F. Banchoff | title = Differential Geometry | author-link = Thomas F. Banchoff | website = Brown University Math Department | access-date = 2024-05-26 | quote = Fenchel's Theorem Theorem: The total curvature of a regular closed space curve C is greater than or equal to 2π.}}
{{cite web | url = https://www.math.brown.edu/tbanchof/balt/ma106/chern22.html | author = Thomas F. Banchoff | title =

2. Curvature and Fenchel's Theorem | author-link = Thomas F. Banchoff | website = Brown University Math Department | access-date = 2024-05-26}}

Category:Theorems in differential geometry

Category:Theorems in plane geometry

Category:Theorems about curves

Category:Curvature (mathematics)