Positivity

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Comparing volumes by concurrent cross-sections of complex lines: a Busemann–Petty type problem

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Abstract

We consider the problem of comparing the volumes of two star bodies in an even-dimensional Euclidean space \({\mathbb {R}}^{2n} = {\mathbb {C}}^n\) by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann–Petty type theorem holds. Quaternionic and octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen’s inequality. Along the way we provide a criterion that detects which centered bodies are circular. i.e., stabilized by multiplication by complex numbers of unit modulus. Our goal is to present a Busemann–Petty type result with a minimum of required background (in the spirit of L. K. Hua’s book on the classical domains) and, in addition, to suggest characterizations of classes of star bodies by means of integral geometric inequalities.

Keywords

Busemann–Petty problem Complex cross-sections Star body Circular domain Volume characterization 

Mathematics Subject Classification

52A20 52A38 52A40 45A60 

Notes

Acknowledgements

The author wishes to thank Susanna Dann, David Feldman, Daniel Klain, Erwin Lutwak, Mehmet Orhon, Larry Zalcman and others for helpful discussions and the referees for suggesting several improvements in the manuscript.

References

  1. 1.
    Barthe, F., Fradelizi, M., Maurey, B.: A short solution to the Busemann–Petty problem. Positivity 3, 95–100 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Busemann, H., Petty, C.M.: Problems on convex bodies. Math. Scand. 4, 88–94 (1956)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dann, S.: The Busemann–Petty problem in the complex hyperbolic space. Math. Proc. Camb. Philos. Soc. 155(1), 155–172 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dann, S., Zymonopoulou, M.: Sections of convex bodies with symmetries. Adv. Math. 271, 112–152 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Grinberg, E.: Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies. Math. Ann. 291(1), 75–86 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Groemer, H.: Stability of geometric inequalities. In: Gruber, P.M., Wills, J.M., (eds.) Handbook of Convex Geometry, vol. A, pp. lxvi+735; vol. B: pp. i–lxvi and 737–1438. North-Holland Publishing Co., Amsterdam (1993). ISBN: 0-444-89598-152-06 (52–00)Google Scholar
  8. 8.
    Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (Translations of Mathematical Monographs), vol. 6. American Mathematical Society, Providence (1979)Google Scholar
  9. 9.
    Koldobsky, A., König, H., Zymonopoulou, M.: The complex Busemann–Petty problem on sections of convex bodies. Adv. Math. 218, 352–367 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71(2), 232–261 (1988)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rubin, B.: The lower dimensional Busemann–Petty problem for bodies with the generalized axial symmetry. Isr. J. Math. 173, 213–233 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Rubin, B.: Comparison of volumes of convex bodies in real, complex, and quaternionic spaces. Adv. Math. 225(3), 1461–1498 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rudin, W.: Real and Complex Analysis, Chapter 3, 3rd edn. McGraw-Hill, New York (1986)Google Scholar
  14. 14.
    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)MATHGoogle Scholar
  15. 15.
    Toth, G.: Measures of Symmetry for Convex Sets and Stability. Universitext. Springer, Cham (2015)CrossRefGoogle Scholar
  16. 16.
    Zalcman, L.: Mean values and differential equations. Isr. J. Math. 14, 339–352 (1973)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zymonopoulou, M.: The modified complex Busemann–Petty problem on sections of convex bodies. Positivity 13(4), 717–733 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Massachusetts, BostonBostonUSA

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