Skip to main content
SpringerLink
Log in

Convex Quadrics and Their Characterizations by Means of Plane Sections

  • Conference paper
  • First Online:
Bridging Mathematics, Statistics, Engineering and Technology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 24))

Abstract

Ellipses and ellipsoids form a well-established special class of convex surfaces, primarily due to a wide range of their applications in various mathematical disciplines. The present survey deals with a natural extension of this class to that of convex quadrics. It contains a classification of convex quadrics of the Euclidean space Rn and describes, in terms of plane quadric sections, their various characteristic properties among all convex hypersurfaces of Rn, possibly unbounded.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alonso, J., Martín, P.: Some characterizations of ellipsoids by sections. Discrete Comput. Geom. 31, 643–654 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. J. Convex Anal. 13, 169–175 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Alonso, J., Martín, P.: Convex bodies with sheafs of elliptic sections. II. J. Convex Anal. 14, 1–11 (2007)

    MATH  Google Scholar 

  4. Auerbach, H., Mazur, S., Ulam, S.: Sur une propriété caractéristique de l’ellipsoïde. Monatsh. Math. 42, 45–48 (1935)

    Article  MathSciNet  Google Scholar 

  5. Berger, M.: Geometry. II. Springer, Berlin (1987)

    Google Scholar 

  6. Bianchi, G., Gruber, P. M.: Characterization of ellipsoids. Arch. Math. (Basel) 49, 344–350 (1987)

    Google Scholar 

  7. Blaschke, W.: Kreis und Kugel. Viet, Leipzig (1916)

    MATH  Google Scholar 

  8. Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin (1934).

    MATH  Google Scholar 

  9. Brunn, H.: Über Kurven ohne Wendepunkte. Habilitationschrift. Ackermann, München (1889)

    Google Scholar 

  10. Burton, G. R.: Sections of convex bodies. J. London Math. Soc. 12, 331–336 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Busemann, H.: The Geometry of Geodesics. Academic, New York (1955)

    MATH  Google Scholar 

  12. Carathéodory, C.: Über den Variabilitätsbereich der Koefficienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gruber, P. M., Höbinger, J. H.: Kennzeichnungen von Ellipsoiden mit Anwendungen. In: Fuchssteiner, B., Kulisch, U., Laugwitz, D., Liedl, R. (eds.) Jahrbuch Überblicke Mathematik, pp. 9–29. Bibliographisches Institut, Mannheim (1976)

    Google Scholar 

  14. Heil, E., Martini, H.: Special convex bodies. In: Gruber, P. M., Wills, J. M. (eds.) Handbook of Convex Geometry, pp. 347–385. North-Holland, Amsterdam (1993)

    Google Scholar 

  15. Höbinger, J.: Über einen Satz von Aitchison, Petty und Rogers. Dissertation, Technische Hochschule Wien, Wien (1974)

    Google Scholar 

  16. Klee, V. L.: Some characterizations of convex polyhedra. Acta Math. 102, 79–107 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  17. Kubota, T.: On the theory of closed convex surface. Proc. London Math. Soc. 14, 230–239 (1914)

    MATH  Google Scholar 

  18. Kubota, T.: Über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 277–287 (1914)

    Google Scholar 

  19. Kubota, T.: Einfache Beweise eines Satzes über die konvexe geschlossene Fläche. Sci. Rep. Tôhoku Univ. 3, 235–255 (1914)

    Google Scholar 

  20. Montejano, L., Morales, E.: Variations of classic characterizations of ellipsoids and a short proof of the false centre theorem. Mathematika 54, 35–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nakagawa, S.: On some theorems regarding ellipsoids. Tôhoku Math. J. 8, 11–13 (1915)

    MathSciNet  MATH  Google Scholar 

  22. Nakajima, S.: Über konvexe Kurven und Flächen. Tohôku Math. J. 29, 227–230 (1928)

    MATH  Google Scholar 

  23. Osgood, W. F., Graustein, W. C.: Plane and solid analytic geometry. Macmillan, New York (1942)

    Google Scholar 

  24. Penna, M. A., Patterson, R. R.: Projective Geometry and its Applications to Computer Graphics. Prentice-Hall, Englewood Cliffs, NJ (1986)

    MATH  Google Scholar 

  25. Petty, C. M.: Ellipsoids. In: Gruber, P. M., Wills, J. M. (eds.) Convexity and its Applications, pp. 264–276. Birkhäuser, Basel (1983)

    Google Scholar 

  26. Soltan, V.: Convex bodies with polyhedral midhypersurfaces. Arch. Math. 65, 336–341 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Soltan, V.: Affine diameters of convex-bodies–a survey. Expo. Math. 23, 47–63 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Soltan, V.: Convex solids with planar midsurfaces. Proc. Amer. Math. Soc. 136, 1071–1081 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Soltan, V.: Convex solids with homothetic sections through given points. J. Convex Anal. 16, 473–486 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Soltan, V.: Convex quadrics. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 3, 94–106 (2010)

    MathSciNet  Google Scholar 

  31. Soltan, V.: Convex solids with hyperplanar midsurfaces for restricted families of chords. Bul. Acad. Ştiinţe Repub. Moldova. Mat. 2, 23–40 (2011)

    MathSciNet  Google Scholar 

  32. Tietze, H.: Über Konvexheit im kleinen und im grossen und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen. Math. Z. 28, 697–707 (1928)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, George Mason University, Fairfax, VA, 22030, USA

    Valeriu Soltan

Authors
  1. Valeriu Soltan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valeriu Soltan .

Editor information

Editors and Affiliations

  1. , Dept of Mathematics & Computer Sciences, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Bourama Toni

  2. , Department of Technology, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Keith Williamson

  3. , Department of Engineering, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Nasser Ghariban

  4. , Department of Mathematics & Computer Sci, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Dawit Haile

  5. , Department of Mathematics & Computer Sci, Virginia State University, P.O. Box 9068, Petersburg, 23806, Virginia, USA

    Zhifu Xie

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media New York

About this paper

Cite this paper

Soltan, V. (2012). Convex Quadrics and Their Characterizations by Means of Plane Sections. In: Toni, B., Williamson, K., Ghariban, N., Haile, D., Xie, Z. (eds) Bridging Mathematics, Statistics, Engineering and Technology. Springer Proceedings in Mathematics & Statistics, vol 24. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4559-3_12

Download citation

Publish with us

Policies and ethics

Search

Navigation