Uniqueness theorem for locally antipodal Delaunay sets

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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 2R-cluster. This theorem implies, in particular, a new proof of a theorem stating that a locally antipodal Delaunay set all of whose 2R-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. 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. 2.
    L. Sohncke, “Die regelmässigen ebenen Punktsysteme von unbegrenzter Ausdehnung, ” J. Reine Angew. Math 77, 47–101 (1874).MathSciNetGoogle Scholar
  3. 3.
    A. Schoenflies, Krystallsystemeund Krystallstructur (B.G. Teubner, Leipzig, 1891).Google Scholar
  4. 4.
    L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume (Erste Abh.), ” Math. Ann. 70 (3), 297–336 (1911).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    L. Bieberbach, “Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abh.): Die Gruppen mit einem endlichen Fundamentalbereich, ” Math. Ann. 72 (3), 400–412 (1912).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    B. N. Delone, “Geometry of positive quadratic forms, ” Usp. Mat. Nauk 3, 16–62 (1937).MathSciNetGoogle Scholar
  7. 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)].MathSciNetMATHGoogle Scholar
  8. 8.
    N. P. Dolbilin, “Crystal criterion and locally antipodal Delaunay sets, ” Vestn. Chelyab. Gos. Univ., No. 3, 6–17 (2015).MathSciNetGoogle Scholar
  9. 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)].MathSciNetCrossRefMATHGoogle Scholar
  10. 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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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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