Dennis DeTurck, Herman Gluck, Daniel Pomerleano, and David Shea Vick

The Four Vertex Theorem of global differential geometry says that the curvature of a simple closed plane curve, if not constant, has at least four local extrema. The converse asserts that any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function of a simple closed curve. The authors explain the proof of both. (pp. 192)
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