Skip to main content
SpringerLink
Log in

Functional analytic characterizations of classes of convex bodies

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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

References

  1. Alfsen, E.: Compact convex sets and boundary integrals. Springer, Berlin, 1971

    MATH  Google Scholar 

  2. Berg, L.: Shephard’s approximation theorem for convex bodies and the Milman theorem. Math. Scand.25, 19–24 (1969)

    MathSciNet  MATH  Google Scholar 

  3. Burago, Yu.D., Zalgaller, V.A.: Geometric Inequalities. Springer Verlag, Berlin, 1988

    MATH  Google Scholar 

  4. Busemann, H.: A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat. Acad. Sci. U.S.A.35, 27–31 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  5. Busemann, H.: The isoperimetric problem for Minkowski area. Amer. J. Math.71, 743–762 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  6. Busemann, H.: The foundations of Minkowskian geometry. Comment. Math. Helv.24, 156–187 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  7. Busemann, H.: The geometry of Finsler spaces. Bull. Amer. Math. Soc.56, 5–16 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choquet, G.: Lectures on analysis. Vol. II. Benjamin/Cummings, Reading, MA, 1969

    MATH  Google Scholar 

  9. Gardner, R.J.: Intersection bodies and the Busemann-Petty Problem. Trans. Amer. Math. Soc.342, 435–445 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gardner, R.J.: On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull. Amer. Math. Soc.30, 222–226 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gardner, R.J.: A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math.140, 435–447 (1994)

    Article  MATH  Google Scholar 

  12. Gardner, R.J., Volčič, A.: Tomography of convex and star bodies. Advances in Math.108, 367–399 (1994)

    Article  MATH  Google Scholar 

  13. Goodey, P.: Centrally symmetric convex sets and mixed volumes. Mathematika24, 193–198 (1977)

    Article  MathSciNet  Google Scholar 

  14. Goodey, P., Schneider, R., Weil, W.: Projection functions on higher rank Grassmannians. Operator Theory: Adv. and Appl.77, 75–90 (1995), Birkhäuser, Basel

    MathSciNet  Google Scholar 

  15. Goodey, P., Weil, W.: Centrally symmetric convex bodies and Radon transforms on higher order Grassmannians. Mathematika38, 117–133 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goodey, P., Weil, W.: Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom.35, 675–688 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Goodey, P., Weil, W.: Zonoids and generalizations. Handbook of Convex Geometry (Gruber, P.M., Wills, J.M., eds.), North-Holland, Amsterdam, 1993, pp. 1297–1326

    Google Scholar 

  18. Goodey, P., Zhang, G.: Characterizations and inequalities for zonoids. Jour. London Math. Soc. (1995)

  19. Horváth, J.: Topological vector spaces and distributions, Vol. I. Addison-Wesley, Reading, MA, 1966

    MATH  Google Scholar 

  20. Jameson, G.: Ordered linear spaces. Springer, Berlin, 1970

    MATH  Google Scholar 

  21. Lutwak, E.: On some affine isoperimetric inequalities. J. Differential Geom.23, 1–13 (1986)

    MathSciNet  MATH  Google Scholar 

  22. Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. in Math.71, 232–261 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Phelps, R.R.: Lectures on Choquet’s theorem. Van Nostrand, Princeton, NJ, 1966

    MATH  Google Scholar 

  25. Schneider, R.: Convex bodies: the Brunn-Minkowski theory. Cambridge Univ. Press, Cambridge, 1993

    Book  MATH  Google Scholar 

  26. Schneider, R., Weil, W.: Zonoids and related topics. Convexity and its Applications (Gruber, P.M., Wills, J.M., eds.), Birkhäuser, Basel, 1983, pp. 296–317

    Google Scholar 

  27. Treves, F.: Topological vector spaces, distributions and kernels. Academic Press, New York, 1967

    MATH  Google Scholar 

  28. Weil, W.: Centrally symmetric convex bodies and distributions. Israel J. Math24, 352–367 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Weil, W.: Centrally symmetric convex bodies and distributions II. Israel J. Math32, 173–182 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Weil, W.: On surface area measures of convex bodies. Geom. Dedicata9, 299–306 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Weil, W.: Some characterizations of zonoids. Unpublished manuscript (1981)

  32. Zhang, G.: Intersection bodies and the four-dimensional Busemann-Petty problem. Duke Math.71, 233–240 (1993)

    Google Scholar 

  33. Zhang, G.: Intersection bodies and Busemann-Petty inequalities in ℝ4. Ann. Math.140, 331–346 (1994)

    Article  MATH  Google Scholar 

  34. Zhang, G.: No polytope is an intersection body (1994) (to appear)

  35. Zhang, G.: Centered bodies and dual mixed volumes. Trans. Amer. Math. Soc.345, 777–781 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oklahoma, 73019, Norman, OK, USA

    P. Goodey

  2. Department of Mathematics, Polytechnic University, 11201, Brooklyn, NY, USA

    E. Lutwak

  3. Mathematisches Institut II, Universität Karlsruhe, D-76128, Karlsruhe, Germany

    W. Weil

Authors
  1. P. Goodey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Lutwak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. Weil
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research of the first author was supported in part by NSF grants DMS-9207019 and INT-9123373. The research of the second author was supported in part by NSF grant DMS-9123571.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goodey, P., Lutwak, E. & Weil, W. Functional analytic characterizations of classes of convex bodies. Math Z 222, 363–381 (1996). https://doi.org/10.1007/BF02621871

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02621871

Keywords

Search

Navigation