Skip to main content

Advertisement

SpringerLink
Log in

On a relative isodiametric inequality for centrally symmetric, compact, convex surfaces

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

We consider the problem of dividing a centrally symmetric, compact, convex surface into two regions in such a way that the intrinsic diameter of both regions is as small as possible. We discuss the best upper bound for the ratio between the area of the smallest region (relative area) and the maximal relative intrinsic diameter. We provide necessary and sufficient conditions for attaining the equality sign. As a consequence from these conditions, there are many surfaces for which the equality sign is never attained. We present a complete study of the special case of the cube surface obtaining the best possible upper bound for this surface.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Cerdán, A.: Comparing the relative volume with the relative inradius and the relative width. J. Inequal. Appl. 54542, 1–8 (2006)

    Article  Google Scholar 

  • Cerdán, A., Miori, C., Segura Gomis, S.: On relative isodiametric inequalities. Beitr. Algebra Geom. 45, 595–605 (2004a)

  • Cerdán, A., Schnell, U., Segura Gomis, S.: On relative geometric inequalities. Math. Inequal. Appl. 7, 135–148 (2004b)

  • Cerdán, A., Miori, C., Schnell, U., Segura Gomis, S.: Relative geometric inequalities for compact, convex surfaces. Math. Inequal. Appl. 13, 203–216 (2010)

    MathSciNet  MATH  Google Scholar 

  • Cianchi, A.: On relative isoperimetric inequalities in the plane. Boll. Un. Mat. Ital 7(3), 289–325 (1989)

    MathSciNet  Google Scholar 

  • Cotton, A., Freeman, D., Gnepp, A., Ng, T., Yoder, C.: The isoperimetric problem on some singular surfaces. J. Aust. Math. Soc. 78(2), 167–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Gruber, P.M.: Baire categories in convexity. In: Handbook of convex geometry, vol. B, pp. 1327–1346. North-Holland, Amsterdam (1993)

  • Gruber, P.M.: Convex and discrete geometry. Springer, Berlin (2007)

    MATH  Google Scholar 

  • Martini, H., Soltan, V.: Antipodality properties of finite sets in Euclidean space. Discrete Math. 290(23), 221–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Osserman, R.: The isoperimetric inequality. Bull. Amer. Math. Soc. 84(6), 1182–1238 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Peri, C.: On relative isoperimetric inequalities. In: Conference seminar of mathematics, University of Bari, Bari 279 (2000)

  • Vîlcu, C.: On two conjetures of Steinhaus. Geom. Dedicata 79, 267–275 (2000)

    Google Scholar 

Download references

Acknowledgments

We would like to thank Vicente Miquel and Francisco Carreras for interesting discussions about this problem and helpful remarks.

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Alicante, Campus de San Vicente del Raspeig, 03080 , Alicante, Spain

    A. Cerdán & S. Segura Gomis

  2. Fachbereich Mathematik/Naturwissenschaften, University of Applied Sciences Zittau/Görlitz, 02763 , Zittau, Germany

    U. Schnell

Authors
  1. A. Cerdán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. U. Schnell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Segura Gomis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Cerdán.

Additional information

Research partially supported by MICINN project MTM2009-10418, and Fundación Séneca project 04540/GERM/06. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cerdán, A., Schnell, U. & Gomis, S.S. On a relative isodiametric inequality for centrally symmetric, compact, convex surfaces. Beitr Algebra Geom 54, 277–289 (2013). https://doi.org/10.1007/s13366-012-0123-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-012-0123-5

Keywords

Mathematics Subject Classification

Search

Navigation