Unfoldings of affine convex polytopes

Geometry Topology Working Seminar
Friday, September 21, 2012 - 13:05
1 hour (actually 50 minutes)
Skiles 006
Georgia Tech
A well-known problem in discerte convex geometry, attributed to the Dutch painter Durrer and first formulated by G. C. Shephard, is concerned with whether every convex polyope P in Euclidean 3-space has a simpe net, i.e., whether the surface of P can be isometrically embedded in the Euclidean plane after it has been cut along some spanning tree of its edges. In this talk we show that the answer is yes after an affine transformation. In particular the combinatorial structure of P plays no role in deciding its unfoldability, which settles a question of Croft, Falconer, and Guy. The proof employs a topological lemma which provides a criterion for checking embeddedness of immersed disks.