Skip to main content
SpringerLink
Log in

On the Reverse Dual Loomis–Whitney Inequality

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The dual Loomis–Whitney inequality provides the sharp lower bound for the volume of a convex body in terms of its \((n-1)\)-dimensional coordinate sections. In this paper, some reverse forms of the dual Loomis–Whitney inequality are obtained. In particular, we show that the best universal DLW-constant for origin-symmetric planar convex bodies is 1.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Alonso-Gutiérrez, D., Artstein-Avidan, S., González Merino, B., Jiménez, C.H., Villa, R.: Rogers-Shephard and local Loomis–Whitney type inequalities. arXiv:1706.01499

  2. Balister, P., Bollobás, B.: Projections, entropy and sumsets. Combinatorica 32, 125–141 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ball, K.: Cube slicing in \(\mathbb{R}^n\). Proc. Am. Math. Soc. 97, 465–473 (1986)

    MATH  Google Scholar 

  4. Ball, K.: Shadows of convex bodies. Trans. Am. Math. Soc. 327, 891–901 (1991)

    Article  MathSciNet  Google Scholar 

  5. Bennett, J., Carbery, A., Wright, J.: A non-linear generalizaton of the Loomis–Whitney inequality and applications. Math. Res. Lett. 12, 443–457 (2005)

    Article  MathSciNet  Google Scholar 

  6. Brazitikos, S., Giannopoulos, A., Liakopoulos, D.-M.: Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies. Adv. Geom. 18, 345–354 (2018)

    Article  MathSciNet  Google Scholar 

  7. Brzezinski, P.: Volume estimates for sections of certain convex bodies. Math. Nachr. 286, 1726–1743 (2013)

    Article  MathSciNet  Google Scholar 

  8. Burago, YuD, Zalgaller, V.A.: Geometric Inequalities, Grundlehren der Mathematischen Wissenschaften, vol. 285. Springer, New York (1988)

    Google Scholar 

  9. Campi, S., Gardner, R.J., Gronchi, P.: Reverse and dual Loomis–Whitney-type inequalities. Trans. Am. Math. Soc. 368, 5093–5124 (2016)

    Article  MathSciNet  Google Scholar 

  10. Campi, S., Gritzmann, P., Gronchi, P.: On the reverse Loomis–Whitney inequality. Discrete Comput. Geom. 60, 115–144 (2018)

    Article  MathSciNet  Google Scholar 

  11. Campi, S., Gronchi, P.: Estimates of Loomis–Whitney type for intrinsic volumes. Adv. Appl. Math. 47, 545–561 (2011)

    Article  MathSciNet  Google Scholar 

  12. Fremann, H., Shapira, H.: Determining the minimum-area encasing rectangle for arbitrary closed curve. Commun. ACM. 18, 409–413 (1975)

    Article  MathSciNet  Google Scholar 

  13. Gardner, R.J.: Geometric Tomography. Encyclopedia of Mathematics and Its Applications, 2nd edn. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  14. Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  15. Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (German). Springer, Berlin (1957)

    Book  Google Scholar 

  16. Huang, Q., Li, A.-J.: On the Loomis–Whitney inequality for isotropic measures. Int. Math. Res. Not. 2017, 1641–1652 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Li, A.-J., Huang, Q.: The \(L_p\) Loomis–Whitney inequality. Adv. Appl. Math. 75, 94–115 (2016)

    Article  MathSciNet  Google Scholar 

  18. Li, A.-J., Huang, Q.: The dual Loomis–Whitney inequality. Bull. Lond. Math. Soc. 48, 676–690 (2016)

    Article  MathSciNet  Google Scholar 

  19. Liakopoulos, D.-M.: Reverse Brascamp–Lieb inequality and the dual Bollobás–Thomason inequality. Arch. Math. 112, 293–304 (2019)

    Article  MathSciNet  Google Scholar 

  20. Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc. 55, 961–962 (1949)

    Article  MathSciNet  Google Scholar 

  21. Meyer, M.: A volume inequality concerning sections of convex sets. Bull. Lond. Math. Soc. 20, 151–155 (1988)

    Article  MathSciNet  Google Scholar 

  22. O’Rourke, H.: Finding minimal enclosing boxes. Int. J. Comput. Inf. Sci. 14, 183–199 (1985)

    Article  MathSciNet  Google Scholar 

  23. Petty, C.M.: Isoperimetric problems. In: Proc. Conf. Convexity Combinat. Geom., Univ. Oklahoma, pp. 26–41 (1971)

  24. Pfeffer, W.F.: The Riemann Approach to Integration, Cambridge Tracts in Mathematics, vol. 109. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  25. Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 151. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  26. Webb, S.: Central slices of the regular simplex. Geom. Dedic. 61, 19–28 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the referee for the valuable suggestions and the very careful reading of the original manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City, 454000, China

    You-Ran Feng & Ai-Jun Li

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    Qingzhong Huang

  3. Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Qingzhong Huang

Authors
  1. You-Ran Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingzhong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ai-Jun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ai-Jun Li.

Additional information

Publisher's Note

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

Qingzhong Huang was supported by AARMS and NSFC (No. 11701219). Ai-Jun Li was supported by Key Research Project for Higher Education in Henan Province (No. 17A110022).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, YR., Huang, Q. & Li, AJ. On the Reverse Dual Loomis–Whitney Inequality. Results Math 74, 106 (2019). https://doi.org/10.1007/s00025-019-1029-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1029-4

Keywords

Mathematics Subject Classification

Search

Navigation