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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies. Trans. Am. Math. Soc. 355, 4723–4735 (2003)
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
B. Fleury, Between Paouris concentration inequality and variance conjecture. Ann. Inst. Henri Poincar Probab. Stat. 46(2), 299–312 (2010)
R.J. Gardner, The Brunn-Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)
R.J. Gardner, in Geometric Tomography. Encyclopedia of Mathematics, vol. 58 (Cambridge University Press, London, 2006)
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)
Y. Gordon, M. Meyer, S. Reisner, Zonoids with minimal volume-product – a new proof. Proc. Am. Math. Soc. 104(1), 273–276 (1988)
B. Klartag, A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)
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
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)
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
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
Corresponding author
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-3-642-29849-3_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29848-6
Online ISBN: 978-3-642-29849-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)