Saturday Morning Lecture:
"The Four-Vertex Theorem and its Converse"
University of Pennsylvania
Abstract:
The Four Vertex Theorem, which was proved about a century ago, 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.
The Converse to the Four Vertex Theorem, of much more recent vintage, 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.
I'll discuss these theorems, their interesting histories, and possible
generalizations and extensions.
