Mathematische Annalen

, Volume 352, Issue 1, pp 55–71 | Cite as

Concurrent normals to convex bodies and spaces of Morse functions



It is conjectured that if \({K\subset\mathbb R^n}\) is a convex body, then there exists a point in the interior of K which is the point of concurrency of normals from 2n points on the boundary of K. We present a topological proof of this conjecture in dimension four assuming \({\partial K}\) is C1,1. From the assumption that the conjecture fails for \({K\subset\mathbb R^4}\), we construct a retraction from \({\overline K}\) to \({\partial K}\). We apply the same strategy to the problem for lower n, assuming no regularity on \({\partial K}\), and show that it provides very simple proofs for the cases of two and three dimensions (the dimension three case was first proved by Erhard Heil). A connection between our approach to this problem and the homotopy type of some function spaces is also explored, and some conjectures along those lines are proposed.

Mathematics Subject Classification (2000)

52A20 54H25 


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  1. 1.
    Baues H.J., Ferrario D.L.: Stratified fibre bundles. Forum Math. 16(6), 865–902 (2004). doi:10.1515/form.2004.16.6.865 CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Caffarelli L.A.: Boundary regularity of maps with convex potentials. II. Ann. Math. (2) 144(3), 453–496 (1996). doi:10.2307/2118564 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved problems in geometry. Problem Books in Mathematics. Springer, New York (1991) (Unsolved Problems in Intuitive Mathematics, II)Google Scholar
  4. 4.
    Deo N., Klamkin M.S.: Existence of four concurrent normals to a smooth closed curve. Amer. Math. Mon. 77, 1083–1084 (1970)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Heil E.: Existenz eines 6-Normalenpunktes in einem konvexen Körper. Arch. Math. (Basel) 32(4), 412–416 (1979). doi:10.1007/BF01238519 MATHMathSciNetGoogle Scholar
  6. 6.
    Heil, E.: Korrektur zu: “Existenz eines 6-Normalenpunktes in einem konvexen Körper”. Arch. Math. (Basel) 33(5):496 (1979/1980). doi:10.1007/BF01222791.
  7. 7.
    Heil, E.: Concurrent normals and critical points under weak smoothness assumptions. In: Discrete Geometry and Convexity (New York, 1982). Ann. New York Acad. Sci., vol. 440, pp. 170–178. The Annals of the New York Academy of Sciences, New York (1985)Google Scholar
  8. 8.
    Klein J.R.: Coordinate free Morse theory. Manuscr. Math. 71(3), 283–294 (1991). doi:10.1007/BF02568406 CrossRefMATHGoogle Scholar
  9. 9.
    Kuiper N.H.: Double normals of convex bodies. Israel J. Math. 2, 71–80 (1964)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lusternik, L., Schnirelmann, L.: Méthodes topologiques dans les problèmes variationnels. I. Pt. Espaces à un nombre fini de dimensions. Traduit du russe par J. Kravtchenko. Paris: Hermann & Cie. 51 S., 5 Fig. (1934)Google Scholar
  11. 11.
    Steenrod N.: The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton (1951)Google Scholar
  12. 12.
    Wegner B.: Existence of four concurrent normals to a smooth closed hypersurface of E n. Amer. Math. Mon. 80, 782–785 (1973)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Zamfirescu T.: Points on infinitely many normals to convex surfaces. J. Reine Angew. Math. 350, 183–187 (1984). doi:10.1515/crll.1984.350.183 CrossRefMATHMathSciNetGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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