Skip to main content

On a Loomis–Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies

  • 2221 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2050)

Abstract

For a permutationally invariant unconditional convex body \(K\) in \({\mathbb{R}}^{n}\) we define a finite sequence \({({K}_{j})}_{j=1}^{n}\) of projections of the body K to the space spanned by first j vectors of the standard basis of \({\mathbb{R}}^{n}\). We prove that the sequence of volumes \({(\vert {K}_{j}\vert )}_{j=1}^{n}\) is log-concave.

Keywords

  • Permutation Invariance
  • Convex Bodies
  • Negative Correlation Property
  • Wojtaszczyk
  • Brunn-Minkowski Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies. Trans. Am. Math. Soc. 355, 4723–4735 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. S.G. Bobkov, F.L. Nazarov, in On Convex Bodies and Log-Concave Probability Measures with Unconditional Basis. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1807 (Springer, Berlin, 2003), pp. 53–69

    Google Scholar 

  3. B. Fleury, Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincar Probab. Stat. 46(2), 299–312 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  5. R.J. Gardner, in Geometric Tomography. Encyclopedia of Mathematics, vol. 58 (Cambridge University Press, London, 2006)

    Google Scholar 

  6. A. Giannopoulos, M. Hartzoulaki, G. Paouris, On a local version of the Aleksandrov-Fenchel inequalities for the quermassintegrals of a convex body. Proc. Am. Math. Soc. 130, 2403–2412 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Y. Gordon, M. Meyer, S. Reisner, Zonoids with minimal volume-product – a new proof. Proc. Am. Math. Soc. 104(1), 273–276 (1988)

    MathSciNet  MATH  Google Scholar 

  8. B. Klartag, A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. A.W. Marshall, I. Olkin, F. Proschan, in Monotonicity of Ratios of Means and Other Applications of Majorization, ed. by O. Shisha. Inequalities (Academic, New York, 1967), pp. 177–190

    Google Scholar 

  10. M. Pilipczuk, J.O. Wojtaszczyk, The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball. Positivity 12(3), 421–474 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. J.O. Wojtaszczyk, in The Square Negative Correlation Property for Generalized Orlicz Balls. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1910 (Springer, Berlin, 2007), pp. 305–313

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof. K. Oleszkiewicz for a valuable comment regarding the equality conditions in Theorem 1 as well as Prof. R. Latała for a stimulating discussion. Research of the First named author partially supported by NCN Grant no. 2011/01/N/ST1/01839. Research of the second named author partially supported by NCN Grant no. 2011/01/N/ST1/05960.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Piotr Nayar .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Nayar, P., Tkocz, T. (2012). On a Loomis–Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies. In: Klartag, B., Mendelson, S., Milman, V. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2050. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29849-3_19

Download citation