Rendiconti del Circolo Matematico di Palermo

, Volume 33, Issue 3, pp 436–440 | Cite as

Smooth approximation of convex bodies

  • Rolf Schneider


We describe a general approximation procedure for convex bodies which shows, in particular, that a body of constant width can be approximated, in the Hausdorff metric, by bodies of constant width with analytic boundaries (in fact, with algebraic support functions). Moreover, the approximating bodies have (at least) the same symmetries as the original one.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Berg Ch.,Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk.,37 (1969), no. 6, 64 pp.Google Scholar
  2. [2]
    Firey W. J.,Approximating convex bodies by algebraic ones, Arch. Math.,25 (1974), 424–425.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Gruber P. M.,Approximation of convex bodies, In: «Convexity and Its Applications», ed. by P. M. Gruber and J. M. Wills, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1983, 131–162.Google Scholar
  4. [4]
    Hammer P. C.,Approximation of convex surfaces by algebraic surfaces, Mathematika,10 (1963), 64–71.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Jaglom J. M., Boltjanski W. G.,Konvexe Figuren, VEB Deutsch. Verl. d. Wiss., Berlin, 1956. English translation (by P. J. Kelly and L. P. Walton):Convex Figures, Rinehart and Winston, New York, 1961.Google Scholar
  6. [6]
    Schneider R.,Zu einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z.,101 (1967), 71–82.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Schneider R.,On Steiner points of convex bodies, Israel J. Math.,9 (1971), 241–249.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Schneider R.,Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc.,194 (1974), 53–78.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Tanno S.,C -approximation of continuous ovals of constant width, J. Math. Soc. Japan,28 (1976), 384–395.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Wegner B.,Analytic approximation of continuous ovals of constant width, J. Math. Soc. Japan,29 (1977), 537–540.MATHMathSciNetGoogle Scholar
  11. [11]
    Weil W.,Einschachtelung konvexer Körper, Arch. Math.,26 (1975), 666–669.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 1984

Authors and Affiliations

  • Rolf Schneider
    • 1
  1. 1.Mathematisches InstitutFreiburg i. Br.

Personalised recommendations