This summer I will have a chance to work on a 16-week summer research project under a professor in convex/discrete geometry. I'm a first-year student with a fairly good background for my age and I've been working through a graduate text in convex geometry, "Introduction to Convex Polytopes" by Brondsted. I need to decide with my supervisor what topic to focus on for a research proposal.
In general I am interested in any topics related to properties of polytopes, tesselations of spaces, illumination problems, packing of convex bodies, etc.
Which of these areas would be most accessible for a first-year student to work on (around 8-weeks to study and learn the background of the problem, and around 8-weeks to work on the problem and continue to learn background material). I'm interested in what sub-field of convex geometry/discrete geometry offers the opportunity to work on a problem that would fit within this time schedule with a potential tangible result towards the solution.
- Sphere Packings (kissing numbers, density, voronoi cells of sphere packings, etc.)
- Finite Packings by Translates of Convex Bodies (Hadwiger numbers, touching numbers)
- Illumination Conjecture and Related Problems
- Coverings by Planks and Cylinders (Covering with hyperplanes, etc.)
- Volumes of Sphere Arrangements (Kneser-Poulsen Conjecture, etc.)
- Ball-Polyhedra (rigidity, disk-polygons, spindle convex sets)
- Convex Polytopes (antipodality, supporting hyperplanes, etc.)
Any suggestions as to what sort of area of topics (please be at least as specific as the list I've just mentioned) I might be able to access would be very helpful! So far I have a basic understanding of some topics to do with affine spaces, basic topology, graph theory, knot theory, hyperplanes, and convex sets.