[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Approximation of convex bodies by polytopes

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Abstract

LetC be a convex body ofE d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n 2/(d−1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Betke U.—Wills J. M.,Diophantine approximation of convex bodies, Manuscript (1979).

  2. Blaschke W.,Kreis und Kugel, Leipzig: Göschen 1916, New York: Chelsea 1949, Berlin: de Gruyter 1956.

    MATH  Google Scholar 

  3. Bronstein E. M.,ε-entropy of convex sets ad functions, Sibir Mat. Z.,17 (1976), 508–514 = Siber. Math. J.,17 (1977) 393–398.

    MathSciNet  Google Scholar 

  4. Bronstein E. M.—Ivanov L. D.,The approximation of convex sets by polyedra, Sibir. Mat. Z.,16 (1975), 1110–1112 = Siber. Math. J.,16 (1976), 852–853.

    MathSciNet  Google Scholar 

  5. Carlsson S.—Grenander U.,Statistical approximation of plane convex sets, Skand. Aktuar. Tidskr.3/4 (1967), 113–127.

    MathSciNet  Google Scholar 

  6. Dinghas A.,Über das Verhalten der Entfernung zweier Punktmengen bei gleichzeitiger Symmetrierung derselben, Arch. Math.,8 (1957), 46–51.

    Article  MATH  MathSciNet  Google Scholar 

  7. Dudley R.,Metric entropy of some classes of sets with differentiable boundaries, J. Appr. Th.,10 (1974), 227–236. Corrigendum ibid.,26 (1979), 192–193.

    Article  MATH  MathSciNet  Google Scholar 

  8. Eggleston H G.,Problems in Euclidean space, London: Pergamon Press 1957.

    MATH  Google Scholar 

  9. Eisenhart L. P.,Riemannian Geometry, Princeton: Princeton University Press 1949.

    MATH  Google Scholar 

  10. Fejes Tóth L.,Lagerungen in der Ebene, auf der Kugel und im Raum, Berlin-Göttigen-Heidelberg: Springer 1953, 1972.

    MATH  Google Scholar 

  11. Gruber P. M.,Approximation of convex bodies by polytopes, C. R. Acad. Bulg. Sci.,34 (1981), 621–622.

    MATH  MathSciNet  Google Scholar 

  12. Hadwiger H.,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Berlin-Göttingen-Heidelberg: Springer 1957.

    MATH  Google Scholar 

  13. Holmes R. B.,Geometric functional analysis and its applications, New York-Heidelberg-Berlin: Springer 1975.

    MATH  Google Scholar 

  14. Ivanov R. P.,Approximation of convex n-polygons by means of inscribed (n−1)-polygons. (Bulgar., Engl. Summary)., In: Proc. Conf. Bulgar. Math. Soc., Vidin, 1973, 113–122. Sofia: Bulgar. Akad. Nauk. 1974.

    Google Scholar 

  15. Johnson H. H.—Vogt A.,A geometric method for approximating convex arcs, SIAM J. Appl. Math.,38 (1980), 317–325.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenderov P.,Approximation of plane convex compacta by polygons.

  17. Koutroufiotis D.,On Blaschke's rolling theorem, Arch. Math.,23 (1972), 655–660.

    Article  MATH  MathSciNet  Google Scholar 

  18. Macbeath A. M.,An extremal property of the hypersphere, Proc. Cambridge Philos. Soc.,47 (1951), 245–247.

    Article  MATH  MathSciNet  Google Scholar 

  19. McClure D E.—Vitale R A.,Polygonal approximation of plane convex bodies, J. Math. Anal. Appl.,51 (1975), 326–335.

    Article  MATH  MathSciNet  Google Scholar 

  20. McMullen P.—Shephard G C.,Convex polytopes and the upper bound conjecture, Cambridge: Gambridge University Press 1971.

    MATH  Google Scholar 

  21. Pogorelov A. V.,Extrinsic, geometry of convex surfaces, Moscow: Izdatel' stvo «Nauka» 1969 and Jerusalem: AMS 1973.

    Google Scholar 

  22. Popov V. A.,Approximation of convex sets (Bulgar., Engl. Summary). Bull. Inst. Math. Bulgar. Acad. Sci.,11 (1970), 67–69.

    MATH  Google Scholar 

  23. Rogers C. A.,Hausdorff measures, Cambridge: Cambridge University Press 1970.

    MATH  Google Scholar 

  24. Santalò L. A.,Integral geometry and geometric probability, London: Addison.-Wesley 1976.

    MATH  Google Scholar 

  25. Schneider R.,Zur optimalen Approximation konvexer Hyperflächen durch Polyeder. Math. Annalen,256 (1981), 289–301.

    Article  MATH  Google Scholar 

  26. Shephard R.—Wieacker J. A.,Approximation of convex bodies by polytopes. Bull. London Math. Soc.,13 (1981), 149–156.

    Article  MathSciNet  Google Scholar 

  27. Shephard G. C.—Webster R. J.,Metrics for sets of convex bodies, Mathematika,12 (1965), 73–88.

    Article  MathSciNet  MATH  Google Scholar 

  28. Valentine F. A.,Convex sets, New York: Mc. Graw-Hill 1964 and Mannheim: Bibl. Inst. 1968.

    MATH  Google Scholar 

  29. Wieacker J. A.,Einige Probleme der polyedrischen Approximation, Diplomarbeit: Univ. Freiburg 1978.

  30. Wills J. M.,Symmetrisierung an Unterräumen., Arch. Math.,20 (1969), 169–172.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Analysis, Technische Universität Wien, Gußhausstr. 27, A-1040, Vienna

    Peter M. Gruber & Petar Kenderov

  2. Institute of Math. and Mech, Bulgar. Acad. Sci., P. O. Box 373, BG-1090, Sofia

    Peter M. Gruber & Petar Kenderov

Authors
  1. Peter M. Gruber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petar Kenderov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gruber, P.M., Kenderov, P. Approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo 31, 195–225 (1982). https://doi.org/10.1007/BF02844354

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02844354

Keywords

Search

Navigation