Four-vertex theorem

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A ellipse (red) and its evolute (blue), showing the four vertices of the curve, each vertex corresponds to a cusp on the evolute.

The Four-vertex theorem states that the curvature function of a simple, closed plane curve is either constant or has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex.

The Four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya.^[1] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4-point contact with the curve (in general the osculating circle has only 3-point contact with the curve). The Four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument.^[2]

The converse to the Four-vertex theorem states that any continuous, real-valued function of the circle that has at least two local maxima and two local minima is the curvature function of a simple, closed plane curve. The converse was proved for strictly positive functions in 1971 by Herman Gluck as a special case of a general theorem on pre-assigning the curvature of n-spheres.^[3] The full converse to the Four-vertex theorem was proved by Björn Dahlberg shortly before his death in January, 1998 and published posthumously.^[4] Dahlberg's proof uses a winding number argument which is in some ways reminiscent of the standard topological proof of the Fundamental Theorem of Algebra.

One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points. The 3D generalization is not trivial, in fact, one can show that there exist convex, homogeneous objects with less than 4 balance points, see Gomboc.

[edit] Notes

^ Mukhopadhyaya, S. (1909). "New methods in the geometry of a plane arc". Bull. Calcutta Math. Soc. 1: 21-27.
^ Kneser, Adolf (1912). "Bemerkungen uber die Anzahl der Extrema des Krummung auf geschlossenen Kurven und uber verwandte Fragen in einer nicht eucklidischen Geometrie". Festschrift Heinrich Weber: 170-180, Teubner.
^ Gluck, Herman (1971). "The converse to the four-vertex theorem". L'Enseignement Math. 17: 295-309.
^ Dahlberg, Björn (2005). "The converse of the four vertex theorem". Proc. Amer. Math. Soc. 133 (7): 2131-2135.