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Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings [article]

Ezra Miller, Igor Pak
2003 arXiv   pre-print
This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry.  ...  We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra.  ...  We initiate a systematic investigation of this metric combinatorics of convex polyhedra by proving the existence of polyhedral nonoverlapping unfoldings and analyzing the structure of the cut locus.  ... 
arXiv:math/0312253v1 fatcat:7f7enli7sbam7pfzh4w7l6mkwy

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Ezra Miller, Igor Pak
2006 Discrete & Computational Geometry  
We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra.  ...  We prove that S has a polyhedral nonoverlapping unfolding into R d , so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically  ...  the week-long program on Topological and Geometric Combinatorics (April, 2003) .  ... 
doi:10.1007/s00454-006-1249-0 fatcat:a3w55ohhmfejjkpg23tlqi4upy

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

Ezra Miller, Igor Pak
2008 Discrete & Computational Geometry  
We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra.  ...  We prove that S has a polyhedral nonoverlapping unfolding into R d , so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically  ...  the week-long program on Topological and Geometric Combinatorics (April, 2003) .  ... 
doi:10.1007/s00454-008-9052-3 fatcat:ltrez2irufgupiq63mrjxc4tym

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings [chapter]

Ezra Miller, Igor Pak
Twentieth Anniversary Volume:  
We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra.  ...  We prove that S has a polyhedral nonoverlapping unfolding into R d , so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically  ...  the week-long program on Topological and Geometric Combinatorics (April, 2003) .  ... 
doi:10.1007/978-0-387-87363-3_19 fatcat:jydf5qrdxjhgddtxhxm2yjkn5i

Combinatorial restrictions on cell complexes [article]

Florian Frick, Technische Universität Berlin, Technische Universität Berlin, John M. Sullivan
2015
In the last chapter we will give a conceptually simple proof of the result of Miller and Pak (Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings, Discrete Comput.  ...  Das letzte Kapitel enthält einen einfachen Beweis eines Resultats von Miller und Pak (Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings, Discrete Comput.  ...  It is known to be false for non-convex (but star-shaped) 3-polyhedra with convex faces; see Tarasov [92] and Grünbaum [47] .  ... 
doi:10.14279/depositonce-4545 fatcat:3hvfwbr4lfbbxooinv2ejlryvy