Skip to main content
SpringerLink
Log in

Orlicz mixed quermassintegrals

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy-Kubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.

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. Aleksandrov A D. On the theory of mixed volumes. I: Extensions of certain concepts in the theory of convex bodies. Mat Sb, 1937, 2: 947–972

    Google Scholar 

  2. Böröczky K, Lutwak E, Yang D, et al. The log-Brunn-Minkowski inequality. Adv Math, 2012, 231: 1974–1997

    Article  MATH  MathSciNet  Google Scholar 

  3. Böröczky K, Lutwak E, Yang D, et al. The logarithmic Minkowski problem. J Amer Math Soc, 2013, 26: 831–852

    Article  MATH  MathSciNet  Google Scholar 

  4. Chou K S, Wang X J. The L p-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv Math, 2006, 205: 33–83

    Article  MATH  MathSciNet  Google Scholar 

  5. Fenchel W, Jessen B. Mengenfunktionen und konvexe Körper. Danske Vid Selskab Mat-fys Medd, 1938, 16: 1–31

    Google Scholar 

  6. Firey WJ. p-means of convex bodies. Math Scand, 1962, 10: 17–24

    MATH  MathSciNet  Google Scholar 

  7. Fleury B, Guédon O, Paouris G A. A stability result for mean width of L p-centroid bodies. Adv Math, 2007, 214: 865–877

    Article  MATH  MathSciNet  Google Scholar 

  8. Gardner R J. Geometric Tomography. Cambridge: Cambridge University Press, 2006

    Book  MATH  Google Scholar 

  9. Gardner R J, Hug D, Weil W. The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities. J Differential Geom, in press, 2014

    Google Scholar 

  10. Gruber P M. Convex and Discrete Geometry. Berlin: Springer, 2007

    MATH  Google Scholar 

  11. Haberl C, Lutwak E, Yang D, et al. The even Orlicz Minkowski problem. Adv Math, 2010, 224: 2485–2510

    Article  MATH  MathSciNet  Google Scholar 

  12. He B, Leng G, Li K. Projection problems for symmetric polytopes. Adv Math, 2006, 207: 73–90

    Article  MATH  MathSciNet  Google Scholar 

  13. Ludwig M. General affine surface areas. Adv Math, 2010, 224: 2346–2360

    Article  MATH  MathSciNet  Google Scholar 

  14. Lutwak E. The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38: 131–150

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Lutwak E, Yang D, Zhang G. A new ellipsoid associated with convex bodies. Duke Math J, 2000, 104: 375–390

    Article  MATH  MathSciNet  Google Scholar 

  17. Lutwak E, Yang D, Zhang G. L p affine isoperimetric inequalities. J Differential Geom, 2000, 56: 111–132

    MATH  MathSciNet  Google Scholar 

  18. Lutwak E, Yang D, Zhang G. On the L p-Minkowski problem. Trans Amer Math Soc, 2004, 356: 4359–4370

    Article  MATH  MathSciNet  Google Scholar 

  19. Lutwak E, Yang D, Zhang G. L p John ellipsoids. Proc London Math Soc, 2005, 90: 497–520

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Lutwak E, Yang D, Zhang G. Orlicz centroid bodies. J Differential Geom, 2010, 84: 365–387

    MATH  MathSciNet  Google Scholar 

  22. Paouris G, Werner E. Relative entropy of cone-volumes and L p centroid bodies. Proc London Math Soc, 2012, 104: 253–186

    Article  MATH  MathSciNet  Google Scholar 

  23. Rao M M, Ren Z D. Theory of Orlicz Spaces. New York: Marcel Dekker, 1991

    MATH  Google Scholar 

  24. Ren D L. Topics in Integral Geometry. Singapore: World Scientific, 1994

    MATH  Google Scholar 

  25. Ryabogin D, Zvavitch A. The Fourier transform and Firey projections of convex bodies. Indiana Univ Math J, 2004, 53: 667–682

    Article  MATH  MathSciNet  Google Scholar 

  26. Santaló L A. Integral Geometry and Geometric Probability. Cambridge: Cambridge University Press, 2004

    Book  MATH  Google Scholar 

  27. Schütt C, Werner E M. Surface bodies and p-affine surface area. Adv Math, 2004, 187: 98–145

    Article  MATH  MathSciNet  Google Scholar 

  28. Schneider R. Convex bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press, 1993

    MATH  Google Scholar 

  29. Stancu A. The discrete planar L 0-Minkowski problems. Adv Math, 2002, 167: 160–174

    Article  MATH  MathSciNet  Google Scholar 

  30. Thompson A C. Minkowski Geometry. Cambridge: Cambridge University Press, 1996

    Book  MATH  Google Scholar 

  31. Werner E M, Ye D. New L p affine isoperimetric inequalities. Adv Math, 2008, 218: 762–780

    Article  MATH  MathSciNet  Google Scholar 

  32. Werner E M. Rényi divergence and L p-affine surface area for convex bodies. Adv Math, 2012, 230: 1040–1059

    Article  MATH  MathSciNet  Google Scholar 

  33. Xiong G. Extremum problems for the cone volume functional of convex polytopes. Adv Math, 2010, 225: 3214–3228

    Article  MATH  MathSciNet  Google Scholar 

  34. Yaskin V, Yaskina M. Centroid bodies and comparison of volumes. Indiana Univ Math J, 2006, 55: 1175–1194

    Article  MATH  MathSciNet  Google Scholar 

  35. Zou D, Xiong G. Orlicz-John ellipsoids. Adv Math, in press, 2014

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Ge Xiong & Du Zou

Authors
  1. Ge Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Du Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ge Xiong.

Additional information

Dedicated to Professor Ren De-lin on the Occasion of his 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiong, G., Zou, D. Orlicz mixed quermassintegrals. Sci. China Math. 57, 2549–2562 (2014). https://doi.org/10.1007/s11425-014-4812-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4812-4

Keywords

MSC(2010)

Search

Navigation