Inventiones mathematicae

, Volume 107, Issue 1, pp 543–560 | Cite as

How to cage an egg

  • Oded Schramm


This paper proves that given a convex polyhedronP ⊂ ℝ3 and a smooth strictly convex bodyK ⊂ ℝ3, there is some convex polyhedronQ combinatorically equivalent toP whichmidscribes K; that is, all the edges ofQ are tangent toK. Furthermore, with some stronger smoothness conditions on ∂K, the space of all suchQ is a six dimensional differentiable manifold.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [An1] Andreev, E.M.: On convex polyhedra in Lobačevskiî spaces. Mat. Sb., Nov. Ser.81 (123), 445–478 (1970); English translation in Math. USSR, Sb.10, 413–440 (1970)Google Scholar
  2. [An2] Andreev, E.M.: On convex polyhedra of finite volume in Lobaçevskiî space. Mat. Sb., Nov. Ser.83 (125), 256–260 (1970); English translation in Math. USSR, Sb.12, 255–259 (1970)Google Scholar
  3. [Ca] Cauchy, A.L.: Sur les polygones et polyèdres, second Mémoire. J. Éc. Polytechnique19, 87–98 (1813)Google Scholar
  4. [Eg] Eggleston, H.G.: Convexity, Cambridge: Cambridge University Press 1958Google Scholar
  5. [Gr] Grünbaum, B.: Convex Polytopes. New York: Wiley 1967Google Scholar
  6. [G-S] Grünbaum, B., Shephard, G.: Some problems on polyhedra. J. Geom.29, 182–190 (1987)Google Scholar
  7. [Koe] Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Saechs. Akad. Wiss. leipzig, Math.-Phys. Kl.88, 141–164 (1936)Google Scholar
  8. [Ro] Roth, B.: Rigid and flexible frameworks. Am. Math. Mon.88, 6–21 (1981)Google Scholar
  9. [Schr] Schramm, O.: Existence and uniqueness of packings with specified combinatorics. Isr. J. Math. (to appear)Google Scholar
  10. [Schu] Schulte, E.: Analogues of Steinitz's theorem about non-inscribable polytopes. In: Böröczky, K., Tóth, G.F. (eds.) Intuitive geometry. Siófok, 1985 (Colloq. Math. Soc. János Bolyai, vol. 48, pp. 503–516). Amsterdam: North-Holland 1987Google Scholar
  11. [St] Steinitz, E.: Über isoperimetrische Probleme bei konvexen Polyedern. J. Reine Angew. Math.159, 133–143 (1928)Google Scholar
  12. [Th] Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton University Notes 1982Google Scholar
  13. [Tu] Tutte, W.T.: How to draw a graph. Proc. Lond. Math. Soc.52, 743–767 (1963)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Oded Schramm
    • 1
  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations