Skip to main content
Log in

L p Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this article, some Brunn–Minkowski type inequalities for (radial) Blaschke–Minkowski homomorphisms with respect to (radial) L p Minkowski addition are established.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abardia J., Bernig A.: Projection bodies in complex vector spaces. Adv. Math. 227, 830–846 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alesker S., Bergin A., Schuster F.E.: Harmonic analysis of translation invariant valuations. Geom. Funct. Anal. 21, 751–773 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakelman I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  4. Borell C.: The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30, 202–216 (1975)

    Article  MathSciNet  Google Scholar 

  5. Borell C.: Capacitary inequality of the Brunn–Minkowski inequality type. Math. Ann. 263, 179–184 (1993)

    Article  MathSciNet  Google Scholar 

  6. Burago Y.D., Zalgaller V.A.: Geometric Inequality. Springer, New York (1988)

    Google Scholar 

  7. Fan K.: Some inequality concerning positive-definite hermitian matrices. Proc. Camb. Phil. Soc. 51, 414–421 (1958)

    Article  Google Scholar 

  8. Fan K.: Problem 4786. Am. Math. Monthly 65, 289 (1958)

    Article  Google Scholar 

  9. Fiedler M., Markham T.: Some results on the Bergström and Minkowski inequalities. Linear Algebra Appl. 232, 199–211 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Firey W.J.: p-means of convex bodies. Math. Scand. 10, 17–24 (1962)

    MathSciNet  MATH  Google Scholar 

  11. Gardner R.J.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)

    Article  MATH  Google Scholar 

  12. Gardner R.J.: Geometric Tomography. 2nd edn. Cambridge University Press, NewYork (2006)

    MATH  Google Scholar 

  13. Gardner R.J., Gronchi P.: A Brunn–Minkowski inequality for the integer lattice. Trans. Am. Math. Soc. 353, 3995–4024 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ludwig M.: Projection bodies and valuations. Adv. Math. 172(2), 158–168 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ludwig M.: Intersection bodies and valuations. Am. J. Math. 126, 1409–1428 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Lutwak E.: The Brunn–Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131–150 (1993)

    MathSciNet  MATH  Google Scholar 

  18. Osserman R.: The Brunn–Minkowski inequality for multiplictities. Invent. Math. 125, 405–411 (1996)

    Article  MathSciNet  Google Scholar 

  19. Parapatits L., Schuster F.E.: The Steiner formula for Minkowski valuations. Adv. Math. 230, 978–994 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schneider R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  21. Schuster F.E.: Volume inequalities and additive maps of convex bodies. Mathematika 53(2), 211–234 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schuster F.E.: Valuations and Busemann–Petty type problems. Adv. Math. 21(1), 344–368 (2008)

    Article  MathSciNet  Google Scholar 

  23. Schuster F.E.: Crofton measures and Minkowski valuations. Duke Math. J. 154, 1–30 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schuster, F.E., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. (accepted)

  25. Wannerer, T.: GL(n) equivariant Minkowski valuations. Indiana Univ. Math. J. (accepted)

  26. Zhao C.: On Blaschke–Minkowski homomorphisms. Geom. Dedic. 149, 373–378 (2010)

    Article  MATH  Google Scholar 

  27. Zhao C., Cheung W.S.: Radial Blaschke–Minkowski homomorphisms and volume differences. Geom. Dedic. 154, 81–91 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhao C., Leng G.: Brunn–Minkowski inequality for mixed intersection bodies. J. Math. Anal. Appl. 301, 115–123 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Wang.

Additional information

A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (11C0542) and the National Natural Science Foundations of China (NO. 11071156 and 11101216).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, W. L p Brunn–Minkowski type inequalities for Blaschke–Minkowski homomorphisms. Geom Dedicata 164, 273–285 (2013). https://doi.org/10.1007/s10711-012-9772-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9772-7

Keywords

Mathematics Subject Classification (2000)

Navigation