Homage to Björn Dahlberg

I'm finally finished with my master's thesis. It's examines a "proper" notion of curvature for polygons, for which the four vertex theorem holds. It all started two years ago, when I got interested in solving the Frenet equations for space curves. In particular, to find necessary and sufficient conditions on pair of periodic functions to be curvature and torsion of a closed space curve. Now for a plane curve, these conditions are


But for space curves, this is an open problem. My initial idea was to try to prove the converse of the four vertex theorem on the sphere, use integral formulas for curvature in terms of torsion for spherical curves, to try to prove a converse four flattening theorem for space curves, where the four flattening is a natural generalization of the four vertex theorem for space curves. But it turned out to be harder than I thought, and remains on my todo-list so to speak.
A year passed, doing discrete differential geometry under a Marie Curie scholarship at TU-Berlin. Suddenly I needed a degree, and I decided to take up the four vertex theorem again, as a subject for my thesis. Inspired by the motto of doing "intelligent" discretizations of "classical" concepts, I took on the discrete four vertex theorem.

An interesting side note is that not only did Björn Dahlberg, who worked at my university in Gothenburg and died suddenly ten years ago, prove the full converse four vertex theorem. He also proved the strongest four vertex theorem for polygons! And both these works were published posthumously (and are just now starting to get attention).

Note, a version of the thesis more suitable for making a booklet is available here.