The University of Michigan Combinatorics Seminar
Everyone knows from their childhood school days how to glue a flat piece of paper to make a convex polyhedral sphere---that is, the boundary of a 3-polytope. Every 3-polytope has such a "foldout", although to this day the only known proofs allow cuts through interiors of facets. This talk is about higher dimensional generalizations: the boundary of every (d+1)-polytope has a non-overlapping foldout in Euclidean d-space, allowing cuts through interiors of facets. Along with the proof comes an algorithm for constructing the foldout, which raises fundamental questions about the computational complexity of convex polyhedral spheres as metric spaces. This work is joint with Igor Pak.