(Georgia Institute of Technology, 2023-07-28)
Barvinok, Nicholas
A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable.
The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability criterion for almost flat convex caps due to Tarasov.
Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Durer's problem is false for pseudo-edge unfoldings. We then use the Maxwell-Cremona Correspondence to present evidence both for and against Durer's problem.