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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.


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closed as too localized by J. M., Potato, Danny Cheuk, Micah, Quixotic Jun 16 '13 at 5:34

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What your supervisor suggests, after some discussion. –  André Nicolas Jan 28 '12 at 2:32
I'm sure that it is not an unfair question to ask what areas of discrete geometry are more accessible than others? –  Samuel Reid Jan 28 '12 at 2:46
It is certainly a fair question. But most areas of mathematics have accessible problems and not so accessible problems. Your supervisor will know intimately about accessible problems in her/his general research area, and will be able to provide assistance when you have difficulties. Congratulations on having earned the opportunity to do supervised research so early! –  André Nicolas Jan 28 '12 at 2:53
Günter Ziegler's Lectures on Polytopes brings you closer to the forefront of research than does Brøndsted, in my opinion. –  Joseph O'Rourke Jan 30 '12 at 0:38
@Samuel: I don't think your question is answerable, as it depends upon personal preferences. They are all accessible; they are all inaccessible. Choose whichever topic appeals to you most aesthetically, and the attraction will fuel your research. –  Joseph O'Rourke Jan 30 '12 at 15:27