Skip to main content
SpringerLink
Log in

L p -mixed intersection bodies

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper the author first introduce a new concept of L p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L p intersection body to L p -mixed intersection body. Inequalities for L p -dual mixed volumes of L p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities.

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

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Goodey P, Lutwak E, Weil W. Functional analytic characterizations of classes of convex bodies. Math Z, 222: 363–381 (1996)

    MATH  MathSciNet  Google Scholar 

  3. Zhang G. A positive solution to the Busemann-Petty problem in ℝ4. Ann Math, 149: 535–543 (1999)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Gardner R J, Koldobsky A, Schlumprecht T. An analytic solution to the Busemann-Petty problem on sections of convex bodies. Ann Math, 149: 691–703 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Busemann H. Volume in terms of concurrent cross-sections. Pacific J Math, 3: 1–12 (1953)

    MATH  MathSciNet  Google Scholar 

  7. Campi S. Stability estimates for star bodies in terms of their intersection bodies. Mathematika, 45: 287–303 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Campi S. Convex intersection bodies in three and four dimensions. Mathematika, 46: 15–27 (1999)

    MATH  MathSciNet  Google Scholar 

  9. Fallert H, Goodey P, Weil W. Spherical projections and centrally symmetric sets. Adv Math, 129: 301–322 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Gardner R J. Geometric Tomography. Cambridge: Cambridge University Press, 1995

    MATH  Google Scholar 

  14. Gardner R J, Koldobsky A, Schlumprecht T. An analytic solution to the Busemann-Petty problem. C R Acad Sci Paris S I Math, 328: 29–34 (1999)

    MATH  MathSciNet  Google Scholar 

  15. Goodey P, Weil W. Intersection bodies and ellipsoids. Mathematika, 42: 295–304 (1995)

    MATH  MathSciNet  Google Scholar 

  16. Grinberg E, Zhang G. Convolutions, transforms, and convex bodies. Proc London Math Soc, 78: 77–115 (1999)

    Article  MathSciNet  Google Scholar 

  17. Koldobsky A. A functional analytic approach to intersection bodies. Geom Funct Anal, 10: 1507–1526 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koldobsky A. Intersection bodies, positive definite distributions, and the Busemann-Petty problem. Amer J Math, 120: 827–840 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Koldobsky A. Second derivative test for intersection bodies. Adv Math, 136: 15–25 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Koldobsky A. Intersection bodies in R4. Adv Math, 136: 1–14 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Koldobsky A. Intersection bodies and the Busemann-Petty problem. C R Acad Sci Paris S I Math, 325: 1181–1186 (1997)

    MATH  MathSciNet  Google Scholar 

  22. Ludwig M. Intersection bodies and valuations. Amer J Math, 128: 1409–1428 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Moszyńska M. Quotient star bodies, intersection bodies, and star duality. J Math Anal Appl, 232: 45–60 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang G. Intersection bodies and polytopes. Mathematika, 46: 29–34 (1999)

    MATH  MathSciNet  Google Scholar 

  25. Zhang G. Sections of convex bodies. Amer J Math, 118: 319–340 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  27. Zhang G. Centered bodies and dual mixed volumes. Trans Amer Math Soc, 345: 777–801 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang G. Intersection bodies and the four-dimensional Busemann-Petty problem. Internat Math Res Notices, 7: 233–240 (1993)

    Article  Google Scholar 

  29. Lutwak E. Mixed projection inequalities. Trans Amer Math Soc, 287: 91–105 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lutwak E. Inequalities for mixed projection bodies. Trans Amer Math Soc, 339: 901–916 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lutwak E. Volume of mixed bodies. Trans Amer Math Soc, 294: 487–500 (1986)

    Article  MathSciNet  Google Scholar 

  32. Lutwak E. Dual mixed volumes. Pacific J Math, 58: 531–538 (1975)

    MATH  MathSciNet  Google Scholar 

  33. Haberl C, Ludwig M. A characterization of Lp intersection bodies. Internat Math Res Notices, (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou, 310018, China

    ChangJian Zhao

Authors
  1. ChangJian Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ChangJian Zhao.

Additional information

This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, C. L p -mixed intersection bodies. Sci. China Ser. A-Math. 51, 2172–2188 (2008). https://doi.org/10.1007/s11425-008-0074-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0074-3

Keywords

MSC(2000)

Search

Navigation