Skip to main content
SpringerLink
Log in

Some comments on floating and centroid bodies in the plane

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \(K_w\) be the wet part of K. In this paper we study some properties of the locus of the centroid of \(K_w\) and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auerbach, H.: Sur un problème de M. Ulam concernant l’equilibre des corps flottants, Studia Math. 7, 121–142 (1938)

    MATH  Google Scholar 

  2. Bracho, J., Montejano, L., Oliveros, D.: A classification theorem for Zindler carrousels. J. Dyn. Control Syst. 7, 367–384 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bracho, J., Montejano, L., Oliveros, D.: Carousels, Zindler curves and the floating body problem. Period. Math. Hungar. 49, 9–23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Flanders, H.: Solution to problem 4828. Am. Math. Mon. 67, 809–810 (1960)

    Article  Google Scholar 

  5. Grünbaum, B.: Measures of symmetry for convex sets. In: Proceedings of Symposia Pure Mathematics, vol. VII, pp. 233–270 (1963)

  6. Harazis̆vili, A.B.: Some properties of the center of symmetry. Soobsch. Akad. Nauk. Gruzin SSR 89, 21–24 (1978)

    MathSciNet  Google Scholar 

  7. Ivaki, M.N.: The planar Busemann–Petty centroid inequality and its stability. Trans. Am. Math. Soc. 368(5), 3539–3563 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jerónimo-Castro, J.: A characterization of the ellipse related to illumination bodies. Elem. Math. 70, 95–102 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jerónimo-Castro, J., Tabchnikov, S.: Configurations spaces of plane polygons and a sub-Riemannian approach to the equitangent problem. J. Dyn. Control Syst. 22, 227–250 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Klamkin, M.S.: Problem 4828. Am. Math. Mon. 66, 816 (1959)

    Article  Google Scholar 

  11. Odani, K.: On Ulam’s floating body problem of two dimension. Bull. Aichi Univ. Educ. 58, 1–4 (2009). (Natural Sciences)

    Google Scholar 

  12. Petty, C.M.: Centroid surfaces. Pacific J. Math. 11, 1535–1547 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  13. Schütt, C., Werner, E.: The convex floating body. Math. Scand. 66, 275–290 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tabachnikov, S.: Tire track geometry: variations on a theme. Israel J. Math. 151, 1–28 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Toponogov, V.A.: Differential Geometry of Curves and Surfaces, A Concise Guide. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  16. Mauldin, R.D.: The Scottish Book. Mathematics from the Scottish Café. Birkhauser, Boston (1981)

    MATH  Google Scholar 

  17. Valentine, F.A.: Convex Sets. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  18. Várkonyi, P.L.: Floating body problems in two dimensions. Stud. Appl. Math. 122, 195–218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wegner, F.: Floating bodies of equilibrium. Stud. Appl. Math. 111, 167–183 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wegner, F.: Floating bodies of equilibrium in 2D, the tire track problem and electrons in an homogeneous magnetic field, e-print archive physics/0203061 (2007)

Download references

Author information

Authors and Affiliations

  1. Facultad de Ingeniería, Universidad Autónoma de Querétaro, Cerro de las Campanas s/n, C.P. 76010, Querétaro, Mexico

    Z. Guerrero-Zarazua & J. Jerónimo-Castro

Authors
  1. Z. Guerrero-Zarazua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Jerónimo-Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Jerónimo-Castro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guerrero-Zarazua, Z., Jerónimo-Castro, J. Some comments on floating and centroid bodies in the plane. Aequat. Math. 92, 211–222 (2018). https://doi.org/10.1007/s00010-017-0525-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-017-0525-4

Mathematics Subject Classification

Search

Navigation