Goal: Consider a unit area convex polygon P in the plane. Pick a point x on the boundary of P and “zip” the boundary of P together starting at x. You can try this with tape and a piece of paper. By a theorem of Alexandrov , the 2-dimensional surface that you obtain can be realized (isometrically) as the boundary of a unique convex polyhedron C in R^3. The combinatorics of C will change as we vary x. In particular, the space of possible polyhedra C will be of interest. In this project, we would like to understand which values of x for different shapes P will maximize this volume. The complete answer to this question is only known when P is a square. For non-convex P, it was also studies for “L-shaped” polygons. We will work on algorithms to compute the shape and visualization of the convex polyhedra obtained by zipping and, if time permits, consider that happens for non-convex P.
Schedule: to be determined.
Persons involved: Advised by Andrew Yarmola.
Difficulty level: Introductory.
Tools: programs in python/sage.
Expected results: Obtain conjectural results for different polygons P and verify known results. Provide proofs for some particular family of polygons.