2022/12/30 by Joseph O’Rourke, O'Rourke, Joseph
Computer Science · Mathematics · #52B10 #52C99 #Advanced Graph Theory Research #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.2 #Mathematics and Applications #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2212.14721
openalex publication_date 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result as establishing that every combinatorial polyhedron has a metric realization that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of ``reverse"): Is there a combinatorial polyhedron such that, for every metric realization P in R3, and for every spanning cut-tree T, P cut by T unfolds to a net? In this note we prove the answer is NO: every combinatorial polyhedron has a realization and a cut-tree that unfolds the polyhedron with overlap.