Israel Journal of Mathematics

, Volume 3, Issue 2, pp 104–112 | Cite as

Finite sets on curves and surfaces

  • H. Guggenheimer


A complete proof is given for Schnirelmann’s theorem on the existence of a square inC 2 Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR n,C 3-diffeomorphic toS n−1, there exist 2n points which are the vertices of a regular 2 n -cellC n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC 2 curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ).


Line Element Bounded Variation Jordan Curve Plane Curf Main Lemma 
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  1. 1.
    C. Carathéodory,Die Kurven mit baschränkten Biegungen. Sitz. Ber. Preuss. Akad. Wiss. Berlin, Math. Phys. Kl., (1933), 102–125. (Gesammelte Werke, vol. II)Google Scholar
  2. 2.
    G. Salmon and W. Fiedler,Analitische Geometrie des Raumes, 2 Aufl. B. G. Teubner, Leipzig, (1874); sec. 95.Google Scholar
  3. 3.
    H. Whitney,Differentiable manifolds. Ann. of Math. (2)37 645–680, 1936; and remark by S. S. Chern, La géometrie des sous-variétés d’un espace euclidien à plusieurs dimensions,Eus. Math. 40 26–46, 1954.CrossRefMathSciNetGoogle Scholar
  4. 4.
    L. G. Šnirel’man,O nekotoryh geometričeskih svoistvah zamknutyh krivyh. Sbornik rabot matematičeskogo razdela sekcii estestvennyh i točnyh nauk Komakademii, Moskva 1929. Reproduced inUspehi Matematičeskih Nauk,10, (1944), 34–44.Google Scholar

Copyright information

© Hebrew University 1965

Authors and Affiliations

  • H. Guggenheimer
    • 1
  1. 1.University of MinnesotaMinneapolis

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