Skip to main content

Advertisement

Log in

The Dual Brunn–Minkowski Inequalities in Spherical and Hyperbolic Spaces

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

The dual Brunn–Minkowski inequalities on star bodies in spherical and hyperbolic spaces have been established, together with precise equality conditions.

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. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)

    Book  Google Scholar 

  2. Cordero-Erausquin, D.: Inégalité de Prékopa-Leindler sur la sphére. C. R. Acad. Sci. Paris Sér. I Math. 329, 789–792 (1999)

  3. Cordero-Erausquin, D., McCann, R. J., Schmuckenschläger, M.: A Riemannian interpolation inequality ä la Borell, Brascamp and Lieb. Invent. Math. 146, 219–257 (2001)

  4. Dan, S., Kim, J., Yaskin, V.: Busemann’s intersection inequality in hyperbolic and spherical spaces. Adv. Math. 326, 521–560 (2018)

    Article  MathSciNet  Google Scholar 

  5. Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge University Press, New York (2006)

    Book  Google Scholar 

  6. Gardner, R.J.: The dual Brunn-Minkowski theory for bounded borel sets: dual affine quermassintergrals and inequalities. Adv. Math. 216, 358–386 (2007)

    Article  MathSciNet  Google Scholar 

  7. Gardner, R. J., Vol\(\check{c}\)i\(\check{c}\), A. Tomography of convex and star bodies. Adv. Math. 108, 367–399 (1994)

  8. Hardy, G. H., Littlewood, J. E., P\(\acute{o}\)lya, G.: Inequalities. Cambridge University Press, Cambridge (1959)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Lutwak, E.: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problems. J. Differ. Geom. 38, 131–150 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Lutwak, E.: The Brunn-Minkowski-Firey theory II. Adv. Math. 118, 244–294 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Xia, W.Y.: Dual Orlicz mixed affine quermassintegrals. Results Math. 12, 1863–1695 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Xiong, G., Zou, D.: Orlicz mixed quermassintegrals. Sci. China Math. 57, 2549–2562 (2014)

  19. Yaskin, V.: The Busemann-Petty problem in hyperbolic and spherical spaces. Adv. Math. 203, 537–553 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yunwei Xia.

Additional information

Communicated by M.Reza Koushesh.

Publisher's Note

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

Supported by Fundamental Research Funds for the Central Universities (XDJK2017B017)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, Y. The Dual Brunn–Minkowski Inequalities in Spherical and Hyperbolic Spaces. Bull. Iran. Math. Soc. 48, 2843–2854 (2022). https://doi.org/10.1007/s41980-021-00670-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-021-00670-z

Keywords

Mathematics Subject Classification

Navigation