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Journal of Mathematical Sciences

, Volume 140, Issue 4, pp 511–527 | Cite as

An isoperimetric problem for tetrahedra

  • V. A. Zalgaller
Article

Abstract

It is proved that a regular tetrahedron has the maximal possible surface area among all tetrahedra having surface with unit geodesic diameter. An independent proof of O’Rourke-Schevon’s theorem about polar points on a convex polyhedron is given. A. D. Aleksandrov’s general problem on the area of a convex surface with fixed geodesic diameter is discussed. Bibliography: 4 titles.

Keywords

Short Path Convex Polyhedron Unique Minimum Special Pair Regular Tetrahedron 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W. Blaschke, Kreis und Kugel [in German], Chelsea, reprint (1949).Google Scholar
  2. 2.
    S. T. Yau, “Problem section,” in: Seminar of Differential Geometry (Ann. of Math. Studies 102), Princeton Univ. Press, Princeton (1982), pp. 669–706.Google Scholar
  3. 3.
    J. O’Rourke and C. A. Schevon, Preprint 27708-0129 Duke Univ., Durham, North Carolina (1993).Google Scholar
  4. 4.
    A. D. Aleksandrov, Convex Polyhedra, Springer Verlag (2005).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • V. A. Zalgaller
    • 1
  1. 1.Weizmann InstituteIsrael

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