Math 2.0

Differential geometry Curves in the plane and in space Theorem

Fenchel's theorem

Builds on Total curvatureCurvature of a curveClosed curveRegular curveConvex set

Statement

Let γ ⁣:[a,b]→ℝ3\gamma \colon [a, b] \to \R^3 be a regular closed curve of class C2C^2 whose derivatives up to order 22 also agree at aa and bb. Then the total curvature of γ\gamma satisfies ∫abκ(t) ∥γ′(t)∥ dt≥2π.\int_a^b \kappa(t) \, \norm{\gamma'(t)} \, dt \ge 2\pi. Equality holds if and only if γ\gamma is a simple closed curve that lies in a plane and bounds a convex region of that plane.

Remarks

The unit tangent T=γ′/∥γ′∥T = \gamma' / \norm{\gamma'} is a closed curve on the unit sphere, and its length equals the total curvature. For each unit vector vv, the integral of ⟨T,v⟩\ip{T}{v} over a period is ⟨γ(b)−γ(a),v⟩=0\ip{\gamma(b) - \gamma(a)}{v} = 0. So TT is not contained in any open hemisphere, and a closed curve on the sphere with this property has length at least 2π2\pi. Example: a circle has total curvature exactly 2π2\pi, and an ellipse too, since both are plane convex curves. A figure-eight curve in the plane has total curvature greater than 2π2\pi, since it is not simple. The theorem holds in the same form in ℝn\R^n for each n≥2n \ge 2. Werner Fenchel proved it in 1929.

Going deeper

For a student

The proof turns a statement about curves in space into a statement about curves on the sphere, through the tangent indicatrix TT. This trick, the Gauss map of a curve, is the one-dimensional model of the Gauss map of a surface. The Fáry–Milnor theorem sharpens Fenchel's bound for knotted curves, from 2π2\pi to more than 4π4\pi.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Affine combination, concept, no evidence levelAffine combinationconceptCurvature of a curve, concept, no evidence levelCurvature of a curveconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal curvature, concept, no evidence levelTotal curvatureconceptConvex set, definition, no evidence levelConvex setdefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionFenchel's theorem, statement, no evidence levelFenchel's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Affine combination concept
  2. Curvature of a curve concept
  3. Euclidean space object
  4. Function concept
  5. Total curvature concept
  6. Convex set definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Limit of a function of several variables concept
  11. Continuity in R^n definition
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Differentiable on an open set definition
  15. Higher partial derivatives definition
  16. Iterated derivative definition
  17. C^k function characterised by continuous derivatives statement
  18. Parametrised curve definition
  19. Closed curve definition
  20. Tangent vector of a curve definition
  21. Regular curve definition
  22. Fenchel's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Fenchel%27s_theorem
About this page
TagF0BJ2BHSZG
URIm2://F0BJ2BHSZG
Kindstatement, theorem
WritingW0 stub
MSC53A04
StatusThis copy of the site has no build data. Run m2 status F0BJ2BHSZG.
AboutTotal curvature
UsesTotal curvature, Curvature of a curve, Closed curve, Regular curve, Convex set
Same as
Changed2 November 2026

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