The four-vertex theorem states that the curvature function of a simple, closed plane curve has at least four local extrema. 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 in 1909 by Syamadas Mukhopadhyaya.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 4th-order contact with the curve. The four- vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument.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.

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