Pseudo-edge unfoldings of convex polyhedra

Nicholas Barvinok, Mohammad Ghomi

Published 2017-09-14Version 1

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 E with respect to which K is not unfoldable. Thus Durer's conjecture does not hold for pseudo-edge unfoldings. The proof is based on an Alexandrov type existence theorem for convex caps with prescribed curvature, and an unfoldability criterion for almost flat convex caps due to Tarasov.