The Four Vertex Theorem and its Converse

TL;DR

This paper reviews the historical development and proof of the Four Vertex Theorem and its converse, culminating in Dahlberg's full proof that any suitable curvature function corresponds to a simple closed plane curve.

Contribution

It provides a self-contained exposition of Dahlberg's complete proof of the converse to the Four Vertex Theorem, resolving a long-standing mathematical question.

Findings

Proof that any continuous curvature function with at least two maxima and minima corresponds to a simple closed curve.

Historical overview of the theorem's development from Mukhopadhyaya to Dahlberg.

Completion of the proof of the full converse to the Four Vertex Theorem.

Abstract

The Four Vertex Theorem, one of the earliest results in global differential geometry, says that a simple closed curve in the plane, other than a circle, must have at least four "vertices", that is, at least four points where the curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya proved this for strictly convex curves in the plane, and in 1912 Adolf Kneser proved it for all simple closed curves in the plane, not just the strictly convex ones. The Converse to the Four Vertex Theorem says that any continuous real-valued function on the circle which has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. In 1971 Herman Gluck proved this for strictly positive preassigned curvature, and in 1997 Bjorn Dahlberg proved the full converse, without the restriction that the curvature be strictly positive. Publication was delayed by Dahlberg's untimely death in January 1998, but his paper was edited afterwards by Vilhelm Adolfsson and Peter Kumlin, and finally appeared in 2005. The work of Dahlberg completes the almost hundred-year-long thread of ideas begun by Mukhopadhyaya, and we take this opportunity to provide a self-contained exposition.