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An isoperimetric problem for tetrahedra

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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.

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References

  1. W. Blaschke, Kreis und Kugel [in German], Chelsea, reprint (1949).

  2. S. T. Yau, “Problem section,” in: Seminar of Differential Geometry (Ann. of Math. Studies 102), Princeton Univ. Press, Princeton (1982), pp. 669–706.

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  3. J. O’Rourke and C. A. Schevon, Preprint 27708-0129 Duke Univ., Durham, North Carolina (1993).

  4. A. D. Aleksandrov, Convex Polyhedra, Springer Verlag (2005).

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Authors and Affiliations

  1. Weizmann Institute, Israel

    V. A. Zalgaller

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  1. V. A. Zalgaller
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 28–55.

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Zalgaller, V.A. An isoperimetric problem for tetrahedra. J Math Sci 140, 511–527 (2007). https://doi.org/10.1007/s10958-007-0431-8

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  • DOI: https://doi.org/10.1007/s10958-007-0431-8

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