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Collapsible polyhedra and median spaces

Author: M. van de Vel
Journal: Proc. Amer. Math. Soc. 126 (1998), 2811-2818
MSC (1991): Primary 57Q99, 52A01; Secondary 05C99
MathSciNet review: 1452832
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Abstract: It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresponding cubical presentation. Some related open problems have been formulated.

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  • 1. H.-J. Bandelt, J. Hedlíková, Median algebras, Discrete Math., 45 (1983), 1-30. MR 84h:06015
  • 2. R. H. Bing, The geometric topology of 3-manifolds, Amer. Math. Soc. Coll. Publ. 40, Amer. Math. Soc., Providence, R.I., (1983), x$+$238 pp. MR 85j:57001
  • 3. G. Birkhoff, S. A. Kiss, A ternary operator in distributive lattices, Bull. Amer. Math. Soc., 53 (1947), 749-752.
  • 4. V. D. Chepoi, A multifacility location problem on median spaces, Discrete Math. Appl., 64 (1) (1996), 1-29. MR 97c:90058
  • 5. V. D. Chepoi, F. F. Dragan, Computing a median point of a simple rectilinear polygon, Inform. Process. Lett., 49 (1994), 281-285.
  • 6. J. R. Isbell, Six Theorems about injective metric spaces, Comment. Math. Helv., 39 (1964), 65-76. MR 32:431
  • 7. J. R. Isbell, Median algebra, Trans. Amer. Math. Soc., 260 (1980), 319-362. MR 81i:06006
  • 8. J.-H. Mai, Y. Tang, An injective metrization for collapsible polyhedra, Proc. Amer. Math. Soc., 88 (1983), 333-337. MR 84g:54036
  • 9. J. van Mill, A. Schrijver, Subbase characterization of compact topological spaces, Gen. Top. Appl., 10 (1979), 183-201. MR 80d:54023
  • 10. J. van Mill, M. van de Vel, Convexity preserving mappings in subbase convexity theory, Proc. Kon. Ned. Akad. Wet., A 81 (1978), 76-90. MR 80f:52014
  • 11. H. M. Mulder, The interval function of a graph, Math. Centre Tracts 132, Mathematisch Centrum, Amsterdam (1980). MR 82h:05045
  • 12. M. Sholander, Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc., 3 (1952), 369-381. MR 14:9b
  • 13. J. R. Stallings, Lectures on polyhedral topology, Tata Institute of Fundamental Research, Bombay (1967). MR 38:6605
  • 14. M. van de Vel, Matching binary convexities, Top. Appl., 16 (1983), 207-235. MR 85f:52026
  • 15. M. van de Vel, Theory of convex structures, Elsevier Science Publishers, Amsterdam (1993), 540+xv pp. MR 95a:52002
  • 16. E. R. Verheul, Modular metric spaces, Report WS-358 (1989).
  • 17. E. R. Verheul, Modular normed spaces, Report WS-365 (1990).
  • 18. E. R. Verheul, Multimedians in metric and normed spaces, CWI tract 91, Centrum voor Wiskunde en Informatika, Amsterdam, Netherlands (1993). MR 94i:54062

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Additional Information

M. van de Vel
Affiliation: Fakulteit Wiskunde en Informatika, Vrije Universiteit, NL-1081 HV Amsterdam, the Netherlands

Keywords: Collapsible polyhedron, convex structure, cubical complex, gate map, injective metric, median graph, median operator, simplicial complex
Received by editor(s): February 22, 1996
Received by editor(s) in revised form: February 5, 1997
Dedicated: This paper is dedicated to the memory of my son Wouter, 1974–1993
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society

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