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Meissner polyhedra

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Abstract

We develop a concrete way to construct bodies of constant width in dimension three. They are constructed from special embeddings of self-dual graphs.

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References

  1. Pankaj K. Agarwal and János Pach, Combinatorial Geometry, John Wiley & Sons (New York, 1995).

  2. Anciaux Henri, Guilfoyle Brendan: On the three-dimensional Blaschke–Lebesgue problem. Proc. Amer. Math. Soc. 139, 1831–1839 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bezdek K., Connelly R.: Pushing disks apart: The Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Bezdek K., Langi Z., Naszódi M., Papez P.: Ball-polyhedra. Discrete Comput. Geom. 38, 201–230 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bezdek K., Naszódi M.: Rigidity of ball-polyhedra in Euclidean 3-space. European J. Combin. 27, 255–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag (Berlin, 1934).

  7. Brown K. Q.: Voronoi diagrams from convex hulls. Inform. Process. Lett. 9, 223–228 (1979)

    Article  MATH  Google Scholar 

  8. G. D. Chakerian and H. Groemer, Convex bodies of constant width, in: Convexity and its Applications, Birkhäuser (Basel, 1983), pp. 49–96.

  9. Eppstein D.: The farthest point Delaunay triangulation minimizes angles. Comput. Geom. 1, 143–148 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grünbaum B.: A proof of Vázsonyi’s conjecture. Bull. Res. Council Israel, Section A. 6, 77–78 (1956)

    MATH  Google Scholar 

  11. E. Heil and H. Martini, Special convex bodies, in: Handbook of Convex Geometry, Vol. A, North-Holland (Amsterdam, 1993), pp. 347–385.

  12. A. Heppes, Beweis einer Vermutung von A. Vázsonyi, Acta Math. Hungar., 7 (1956), 463–466.

  13. Kupitz Y. S., Martini H., Perles M. A.: Ball polytopes and the Vázsonyi problem. Acta Math. Hungar. 126, 99–163 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lachand-Robert T., Oudet E.: Bodies of constant width in arbitrary dimension. Math. Nachr. 280, 740–750 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lovász L.: Self-dual polytopes and the chromatic number of distance graphs on the sphere. Acta Sci. Math. (Szeged) 45, 317–323 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Meissner E.: Über die durch reguläre Polyeder nicht stützbaren Körper. Vierteljahresschr. Naturfor. Ges. Zürich. 63, 544–551 (1918)

    MATH  Google Scholar 

  17. F. Reuleaux, Theoretische kinematik: Grundzüge einer Theorie des Maschinenwesens, vol. 1, F. Vieweg und Sohn (1875).

  18. Sallee G. T.: Reuleaux polytopes. Mathematika 17, 315–323 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shiohama K., Takagi R.: A characterization of a standard torus in \({E^3}\). J. Differential Geom. 4, 477–485 (1970)

    MathSciNet  MATH  Google Scholar 

  20. S. Straszewicz, Sur un problème géométrique de P. Erdős, Bull. Acad. Polon. Sci. Cl. III, 5 (1957), 39–40.

  21. I. M. Yaglom and V. G. Boltyanskii, Convex Figures, Holt, Rinehart and Winston (1961).

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Correspondence to E. Roldán-Pensado.

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This research was supported by CONACYT project 166306 and PAPITT-UNAM project IN112614.

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Montejano, L., Roldán-Pensado, E. Meissner polyhedra. Acta Math. Hungar. 151, 482–494 (2017). https://doi.org/10.1007/s10474-017-0697-3

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