Four-vertex theorem

The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex.

This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion.

The four-vertex theorem was proved for more general curves by Adolf Kneser in 1912 using a projective argument.

Therefore, there is a local minimum of curvature between each pair of tangencies, giving two of the four vertices.

[10][3] 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.

[11] The full converse to the four-vertex theorem was proved by Björn Dahlberg [de] shortly before his death in January 1998, and published posthumously.

[13] One corollary of the theorem is that a homogeneous, planar disk rolling on a horizontal surface under gravity has at least 4 balance points.

That is, in a coherent polygon, is not allowed for the triangle formed by these three vertices to be obtuse with one of the two edges as its longest side.

[15] It is not necessarily the case that the angles at the vertices of a convex polygon have four local extremes.

[17] The stereographic projection from the once-punctured sphere to the plane preserves critical points of geodesic curvature.

Every simple closed space curve which lies on the boundary of a convex body has four vertices.