Advertisement

Israel Journal of Mathematics

, Volume 154, Issue 1, pp 191–207 | Cite as

Modified Busemann-Petty problem on sections of convex bodies

  • A. Koldobsky
  • V. Yaskin
  • M. Yaskina
Article

Abstract

The Busemann-Petty problem asks whether convex origin-symmetric bodies in ℝ n with smaller central hyperplane sections necessarily have smallern-dimensional volume. It is known that the answer is affirmative ifn≤4 and negative ifn≥5. In this article we replace the assumptions of the original Busemann-Petty problem by certain conditions on the volumes of central hyperplane sections so that the answer becomes affirmative in all dimensions.

Keywords

Convex Body Fractional Derivative Homogeneous Function Positive Real Root Section Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BP] H. Busemann and C. M. Petty,Problems on convex bodies, Mathematica Scandinavica4 (1956), 88–94.MATHMathSciNetGoogle Scholar
  2. [BZ] J. Bourgain and Gaoyong Zhang,On a generalization of the Busemann-Petty problem, inConvex Geometric Analysis (Berkeley, CA, 1996), Mathematical Sciences Research Institute Publications,34, Cambridge University Press, Cambridge, 1999, pp. 65–76.Google Scholar
  3. [GKS] R. J. Gardner, A. Koldobsky and T. Schlumprecht,An analytic solution to the Busemann-Petty problem on sections of convex bodies, Annals of Mathematics149 (1999), 691–703.MATHCrossRefMathSciNetGoogle Scholar
  4. [GV] I. M. Gelfand and N. Ya. Vilenkin,Generalized Functions, Vol 4, Applications of Harmonic Analysis, Academic Press, New York, 1964.Google Scholar
  5. [K1] A. Koldobsky,An application, of the Fourier transform to sections of star bodies, Israel Journal of Mathematics106 (1998), 157–164.MATHMathSciNetGoogle Scholar
  6. [K2] A. Koldobsky,A generalization of the Busemann-Petty problem on sections of convex bodies, Israel Journal of Mathematics110 (1999), 75–91.MATHMathSciNetGoogle Scholar
  7. [K3] A. Koldobsky,A functional analytic approach to intersection bodies, Geometric and Functional Analysis,10 (2000), 1507–1526.MATHCrossRefMathSciNetGoogle Scholar
  8. [K4] A. Koldobsky,Comparison of volumes of convex bodies by means of areas of central sections, Advances in Applied Mathematics33 (2004), 728–732.MATHCrossRefMathSciNetGoogle Scholar
  9. [K5] A. Koldobsky,Fourier Analysis in Convex Geometry, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2005.MATHGoogle Scholar
  10. [MP] V. D. Milman and A. Pajor,Isotropic position and intertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, inGeometric Aspects of Functional Analysis (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Mathematics1376, Springer, Heidelberg, 1989, pp. 64–104.CrossRefGoogle Scholar
  11. [P] C. M. Petty,Projection bodies, Proc. Coll. Convexity (Copenhagen 1965), Kobenhavns Univ. Mat. Inst., pp. 234–241.Google Scholar
  12. [RZ] B. Rubin and Gaoyong Zhang,Generalizations of the Busemann-Petty problem for sections of convex bodies, Journal of Functional Analysis213 (2004), 473–501.MATHCrossRefMathSciNetGoogle Scholar
  13. [S] R. Schneider,Zur einem Problem von Shephard uber die Projektionen konvexer Korper, Mathematische Zeitschrift101 (1967), 71–82 (German).MATHCrossRefMathSciNetGoogle Scholar
  14. [Y] V. Yaskin,The Busemann-Petty problem in hyperbolic and spherical spaces, Advances in Mathematics, to appear.Google Scholar
  15. [Zh] Gaoyong Zhang,A positive answer to the Busemann-Petty problem in four dimensions, Annals of Mathematics149 (1999), 535–543.MATHCrossRefMathSciNetGoogle Scholar
  16. [Zv] A. Zvavitch,The Busemann-Petty problem for arbitrary measures, Mathematische Annalen331 (2005), 867–887.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University 2006

Authors and Affiliations

  • A. Koldobsky
    • 1
  • V. Yaskin
    • 1
  • M. Yaskina
    • 1
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

Personalised recommendations