The Four-or-More Vertex Theorem

The four-vertex theorem states that a smooth Jordan curve in the plane has at least four vertices. A vertex is a local maximum or minimum of the curvature. Thus, an ellipse has exactly four vertices, at the ends of the major and minor axes. This theorem is frequently proved, under the additional assumption that the curve is convex, in introductory differential geometry ([2], [5], [6], [7], [13], [16], [21]) as an early instance of a theorem requiring global rather than purely local arguments. The four-vertex theorem (Vierscheitelsatz, Theoreime des quatre sommets) has a long history, starting in 1909 with Mukhopadhaya [18], who stated and proved it for convex curves. There followed a succession of different proofs, generalizations, and analogies (see the References for a sample), including an interesting recent contribution due to Gluck [9], who proved a kind of converse. It is therefore somewhat surprising that the argument presented here seems not only to be new, but also to have a number of advantages over the usual proofs: