Journal of Mathematical Sciences

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

An isoperimetric problem for tetrahedra

  • V. A. Zalgaller


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.


Short Path Convex Polyhedron Unique Minimum Special Pair Regular Tetrahedron 


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  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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