  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.
