Skip to main content

Advertisement

SpringerLink
Log in

Orlicz inequalities for radial Blaschke–Minkowski homomorphisms

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In the paper, our main aim is to generalize Brunn-Mingkowski inequality for the mixed radial Blaschke–Minkowski homomorphisms into the Orlicz space. The fundamental notions and conclusions of the mixed radial Blaschke–Minkowski homomorphisms as well as related Brunn–Minkowski inequality are extended to an Orlicz setting. The new Orlicz–Brun–Minkowski inequality for the mixed radial Blaschke–Minkowski homomorphisms in special case yields the \(L_p\)-Brunn–Minkowski inequality for mixed radial Blaschke–Minkowski homomorphisms, and mixed intersection bodies, respectively.

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. Chen, F., Leng, G.: Orlicz–Brunn–Minkowski inequality for radial Blaschke–Minkowski homomorphisms. Geom. Dedic. 187(1), 137–149 (2017)

    Article  MATH  Google Scholar 

  2. Firey, W.J.: Polar means of convex bodies and a dual to the Brunn–Minkowski theorem. Can. J. Math. 13, 444–453 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gardner, R.J.: Geometric Tomography. Cambridge University Press, New York (1996)

    MATH  Google Scholar 

  4. Gardner, R.J.: A positive answer to the Busemann–Petty problem in three dimensions. Ann. Math. 140(2), 435–447 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gardner, R.J., Hug, D., Weil, W.: The Orlicz–Brunn–Minkowski theory: a general framework, additions, and inequalities. J. Differ. Geom. 97(3), 427–476 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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  MathSciNet  MATH  Google Scholar 

  7. Haberl, C.: \(L_{p}\) intersection bodies. Adv. Math. 217, 2599–2624 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Haberl, C., Lutwak, E., Yang, D., Zhang, G.: The even Orlicz Minkowski problem. Adv. Math. 224, 2485–2510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, J., Yuan, S., Leng, G.: \(L_{p}\)-Blaschke valuations. Trans. Am. Math. Soc. 367, 3169–3187 (2015)

    Article  MATH  Google Scholar 

  10. Lin, Y.: Affine Orlicz Pólya-Szegö principle for log-concave functions. J. Funct. Anal. 273, 3295–3326 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lutwak, E.: Centroid bodies and dual mixed volumes. Proc. Lond. Math. Soc. 60, 365–391 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Lutwak, E.: The Brunn–Minkowski–Firey theory II. Affine and geominimal surface areas. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Lutwak, E., Yang, D., Zhang, G.: Orlicz projection bodies. Adv. Math. 223, 220–242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lutwak, E., Yang, D., Zhang, G.: Orlicz centroid bodies. J. Differ. Geom. 84, 365–387 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Schuster, F.E.: Valuations and Busemann–Petty type problems. Adv. Math. 219, 344–368 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, W.: \(L_p\) Brunn–Minkowski tyoe inequalities for Blaschke–Minkowski homomorphisms. Geom. Dedic. 164, 273–285 (2013)

    Article  MATH  Google Scholar 

  20. Wang, W.: \(L_p\) radial Blaschke–Minkowski homomorphisms. Taiwan. J. Math. 15, 1183–1199 (2011)

    Article  MATH  Google Scholar 

  21. Werner, E.M.: Rényi divergence and \(L_{p}\)-affine surface area for convex bodies. Adv. Math. 230, 1040–1059 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xi, D., Jin, H., Leng, G.: The Orlicz Brunn–Minkwski inequality. Adv. Math. 260, 350–374 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhao, C.-J.: Orlicz–Brunn–Minkowski inequality for radial Blaschke–Minkowski homomorphisms. Quaest. Math. 41, 937–950 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, C.-J.: On radial and polar Blaschke–Minkowski homomorphisms. Proc. Am. Math. Soc. 141, 667–676 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, C.-J.: The Dual Orlicz–Aleksandrov–Fenchel Inequality. arXiv:2002.11112v1 [math.MG]

  26. Zhao, C.-J.: Orlicz dual affine quermassintegrals. Forum Math. 30(4), 929–945 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhu, B., Zhou, J., Xu, W.: Dual Orlicz–Brunn–Minkwski theory. Adv. Math. 264, 700–725 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, G.: The Orlicz centroid inequality for star bodies. Adv. Appl. Math. 48, 432–445 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author expresses his grateful thanks to the reviewer for his (her) wonderful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, China Jiliang University, Hangzhou, 310018, Zhejiang, People’s Republic of China

    Chang-Jian Zhao

Authors
  1. Chang-Jian Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang-Jian Zhao.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research is supported by National Natural Science Foundation of China (11371334, 10971205).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, CJ. Orlicz inequalities for radial Blaschke–Minkowski homomorphisms. Ricerche mat 71, 333–347 (2022). https://doi.org/10.1007/s11587-020-00501-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-020-00501-z

Keywords

Mathematics Subject Classification

Search

Navigation