Skip to main content
Log in

On the isoperimetric inequalities for Reuleaux polygons

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

We give a unifying approach to the Blaschke-Lebesgue Theorem and the Firey-Sallee Theorem on Reuleaux polygons in the Euclidean plane.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. L. Beretta, A. Maxia: Insiemi convessi e orbiformi.Univ. Roma e Ist. Naz. Alta Mat. Rend. Mat. (5) 1 (1940), 1–64.

    Google Scholar 

  2. A.S. Besicovitch: Minimum area of a set of constant width.Proc. Sympos. Pure Math., Vol. VII, 13–14. Amer. Math. Soc, Providence, R.I., 1963.

    Google Scholar 

  3. W. Blaschke: Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts.Math. Annalen 75 (1915), 504–513.

    Google Scholar 

  4. C. Blatter: Über Kurven konstanter Breite.Elem. Math. 36 (1981), 105–115.

    Google Scholar 

  5. B. Braden: The surveyor's area formula.College Math. J. 17 (1986), 326–337.

    Google Scholar 

  6. G.D. Chakerian: Sets of constant width.Pacific J. Math. 19 (1966), 13–21.

    Google Scholar 

  7. G.D. Chakerian, H. Groemer: Convex bodies of constant width. In:Convexity and its Applications, Eds. P.M. Gruber and J.M. Wills, Birkhäuser, Basel 1983, 49–96.

    Google Scholar 

  8. H.G. Eggleston: A proof of Blaschke's theorem on the Reuleaux triangle.Quart. J. Math. Oxford Ser. (2) 3 (1952), 296–297.

    Google Scholar 

  9. H.G.Eggleston:Convexity. Cambridge Univ. Press, 1958.

  10. W.J. Firey: Isoperimetric ratios of Reuleaux polygons.Pacific J. Math. 10 (1960), 823–829.

    Google Scholar 

  11. H. Groemer: Stability theorems for convex domains of constant width.Canad. Math. Bull. 31 (1988), 328–337.

    Google Scholar 

  12. E.Heil, H.Martini: Special convex bodies. In:Handbook of Convex Geometry, Eds. P.M. Gruber and J.M. Wills, Elsevier Publ., 1993, 347–385.

  13. Y.S.Kupitz:Extremal Problems in Combinatorial Geometry. Aarhus Universitet Lecture Notes, Series 53 (1979).

  14. Y.S.Kupitz, H.Martini: Prom intersectors to successors, to appear.

  15. Y.S. Kupitz, H. Martini, B. Wegner: A linear-time construction of Reuleaux polygons.Beiträge zur Algebra und Geometrie 37 (1996), 415–427.

    Google Scholar 

  16. H.Lebesgue: Sur le probléme isopérimètrique et sur les domains de largeur constante.Bull, de la Soc. Math, de France (Comptes Rendues), 1914, 72–76.

  17. H. Lebesgue: Questions de minimum relatives aux courbes orbiformes.J. de Math, pures et appl. 4 (1921), 67–96.

    Google Scholar 

  18. A.E. Mayer: Der Inhalt der Gleichdicke. Abschätzungen für ebene Gleichdicke.Math. Ann. 110 (1934), 97–127.

    Google Scholar 

  19. E. Meissner: Über Punktmengen konstanter Breite.Vierteljahresschr. Naturforsch. Ges. Zürich 56 (1911), 42–50.

    Google Scholar 

  20. G.T. Sallee: Maximal area of Reuleaux polygons.Canad. Math. Bull. 13 (1970), 175–179.

    Google Scholar 

  21. F.A. Valentine:Convex Sets. McGraw-Hill, New York 1964.

    Google Scholar 

  22. I.M. Yaglom, V.G. Boltyanski:Convex Figures. Holt, Rinehart and Winston, New York, 1961.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kupitz, Y.S., Martini, H. On the isoperimetric inequalities for Reuleaux polygons. J Geom 68, 171–191 (2000). https://doi.org/10.1007/BF01221070

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01221070

Keywords

Navigation