Georgia Institute of Technology College of Sciences

Topological methods have had a rich history of use in convex optimization, including for instance the famous Pataki-Barvinok bound on the ranks of solutions to semidefinite programs, which involves the Borsuk-Ulam theorem. We will give two proofs of a similar sort involving the use of some basic homotopy theory. One is a new proof of Brickman's theorem, stating that the image of a sphere into R^2 under a quadratic map is convex, and the other is an original theorem stating that the image of certain matrix groups under linear maps into R^2 is convex. We will also conjecture some higher dimensional analogues.