2003/12/12 by Ezra Miller, Miller, Ezra, Igor Pak +1
Computer Science · Engineering · Mathematics · #52-04 #52B05 #52B11 #52B70 #52C45 #53C22 #68Q17 #68U05 #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #graph theory and CDMA systems #math.CO #math.MG #msc:52-04 #msc:52B05 #msc:52B11 #msc:52B70 #msc:52C45 #msc:53C22 #msc:68Q17 #msc:68U05
paper · pdf · doi:10.48550/arxiv.math/0312253
47 pages; 21 PostScript (.eps) figures, most in color
arxiv created 2003/12/12 · openalex publication_date 2003/12/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into Rd, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in Rd by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.