Uniqueness theorem for locally antipodal Delaunay sets (original) (raw)

Abstract

We prove theorems on locally antipodal Delaunay sets. The main result is the proof of a uniqueness theorem for locally antipodal Delaunay sets with a given 2_R_-cluster. This theorem implies, in particular, a new proof of a theorem stating that a locally antipodal Delaunay set all of whose 2_R_-clusters are equivalent is a regular system, i.e., a Delaunay set on which a crystallographic group acts transitively.

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References

  1. E. S. Fedorov, An Introductionto the Theory of Figures (St. Petersburg, 1885); 2nd ed. (Akad. Nauk SSSR, Moscow, 1953), Classics of Science [in Russian].
    Google Scholar
  2. L. Sohncke, “Die regelmässigen ebenen Punktsysteme von unbegrenzter Ausdehnung, ” J. Reine Angew. Math 77, 47–101 (1874).
    MathSciNet Google Scholar
  3. A. Schoenflies, Krystallsystemeund Krystallstructur (B.G. Teubner, Leipzig, 1891).
    Google Scholar
  4. L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume (Erste Abh.), ” Math. Ann. 70 (3), 297–336 (1911).
    Article MathSciNet MATH Google Scholar
  5. L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abh.): Die Gruppen mit einem endlichen Fundamentalbereich, ” Math. Ann. 72 (3), 400–412 (1912).
    Article MathSciNet MATH Google Scholar
  6. B. N. Delone, “Geometry of positive quadratic forms, ” Usp. Mat. Nauk 3, 16–62 (1937).
    MathSciNet Google Scholar
  7. B. N. Delone, N. P. Dolbilin, M. I. Shtogrin, and R. V. Galiulin, “A local criterion for regularity of a system of points, ” Dokl. Akad. Nauk SSSR 227 (1), 19–21 (1976) [Sov. Math., Dokl. 17 (2), 319–322 (1976)].
    MathSciNet MATH Google Scholar
  8. N. P. Dolbilin, “Crystal criterion and locally antipodal Delaunay sets, ” Vestn. Chelyab. Gos. Univ., No. 3, 6–17 (2015).
    MathSciNet Google Scholar
  9. N. P. Dolbilin and A. N. Magazinov, “Locally antipodal Delaunay sets, ” Usp. Mat. Nauk 70 (5), 179–180 (2015) [Russ. Math. Surv. 70, 958–960 (2015)].
    Article MathSciNet MATH Google Scholar
  10. R. P. Feynman, R. B. Leighton, and M. Sands, TheFeynman Lectures on Physics (Addison Wesley, Reading, MA, 1964), Vol. II, Ch. 30.
    Google Scholar

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia
    N. P. Dolbilin & A. N. Magazinov

Authors

  1. N. P. Dolbilin
  2. A. N. Magazinov

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Correspondence toN. P. Dolbilin.

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Original Russian Text © N.P. Dolbilin, A.N. Magazinov, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 294, pp. 230–236.

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Dolbilin, N.P., Magazinov, A.N. Uniqueness theorem for locally antipodal Delaunay sets.Proc. Steklov Inst. Math. 294, 215–221 (2016). https://doi.org/10.1134/S0081543816060134

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