Local overlaps in special unfoldings of convex polyhedra
It is an old conjecture of \textit{G. C. Shephard} [Math. Proc. Camb. Philos. Soc. 78, 389--403 (1975; Zbl 0312.52011)] that the boundary of a convex polyhedron (\(3\)-polytope) can be cut up along a spanning tree of its edges and unfolded into the plane without overlaps. In general, unfoldings will have overlaps. Call such an overlap \(k\)-local if there are at most \(k\) vertices in the shortest edge-path between the two overlapping faces. In this paper, the author develops conditions for the occurrence of the minimal case \(k = 2\); as the name suggests, these conditions are purely local. However, the author then applies his results to normal unfoldings, determined by the (increasing, say) angles between the normals to the faces and a fixed direction, giving examples of polyhedra for which all normal unfoldings have overlaps.
scientific article; zbMATH DE number 5566430