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

2003
*
arXiv
*
pre-print

This paper is a study

arXiv:math/0312253v1
fatcat:7f7enli7sbam7pfzh4w7l6mkwy
*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. ...##
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Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

2006
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Discrete & Computational Geometry
*

We present conjectures concerning the number

doi:10.1007/s00454-006-1249-0
fatcat:a3w55ohhmfejjkpg23tlqi4upy
*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) . ...##
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Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

2008
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Discrete & Computational Geometry
*

We present conjectures concerning the number

doi:10.1007/s00454-008-9052-3
fatcat:ltrez2irufgupiq63mrjxc4tym
*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) . ...##
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Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings
[chapter]

*
Twentieth Anniversary Volume:
*

We present conjectures concerning the number

doi:10.1007/978-0-387-87363-3_19
fatcat:jydf5qrdxjhgddtxhxm2yjkn5i
*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) . ...##
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Combinatorial restrictions on cell complexes
[article]

2015

In the last chapter we will give a conceptually simple proof

doi:10.14279/depositonce-4545
fatcat:3hvfwbr4lfbbxooinv2ejlryvy
*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] . ...