Abstract
We study the asymptotic behavior, as the dimension goes to infinity, of the volume of sections of the unit balls of the spaces ℓ n q , 0 < q ≤ ∞. We compute the precise asymptotics of the average volume of central sections and then prove a concentration inequality of exponential type. For the case of non-central hyperplane sections of the cube, we prove a local limit theorem confirming the conjecture on the asymptotically Gaussian dependence of the volume of sections on the distance from the hyperplane to the origin. Note that a weak limit theorem was established very recently in [ABP] for a larger class of bodies. Our calculations are based on connections between volume and the Fourier transform.
The first named author was supported in part by the NSF Grant DMS-9996431. The second named author was supported in parts by RFBR and INTAS Grant 99-01-00112.
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
Anttila M., Ball K., Perissinaki I. The central limit problem for convex bodies. Preprint
Artin E. (1964) The Gamma Function. Athena Series: Selected Topics in Mathematics. Holt, Rinehart and Winston, New York-Toronto-London
Ball K. (1986) Cube slicing in ℝn. Proc. Amer. Math. Soc. 97:465–473
Ball K. (1988) Logarithmically concave functions and sections of convex sets in ℝn. Studia Math. 87:69–84
Ball K. (1989) Volumes of sections of cubes and related problems. Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., 1376, Springer, Berlin-New York, 251–260
Brehm U., Voigt J. Asymptotics of cross sections for convex bodies. Preprint
Diaconis P., Freedman D. (1987) A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincare 23:397–423
Hensley D. (1979) Slicing the cube in ℝn and probability bounds for the measure of a central cube slice in ℝn by probability methods. Proc. Amer. Math. Soc. 73:95–100
Koldobsky A. (1998) An application of the Fourier transform to sections of star bodies. Israel J. Math. 106:157–164
Koldobsky A. (1998) Intersection bodies in ℝ4. Adv. Math. 136:1–14
Koldobsky A. (1999) A generalization of the Busemann-Petty problem on sections of convex bodies. Israel J. Math. 110:75–91
Koldobsky A. A functional analytic approach to intersection bodies. GAFA, to appear
Laplace P.S. (1812) Th'eorie analytique des probabilit'es, Paris
Ledoux M. (1996) Isoperimetry and Gaussian analysis. Lecture Notes Math. 1648:165–296
Lutwak E. (1975) Dual cross-sectional measures. Rend. Acad. Naz. Lincei 58:1–5
Meyer M., Pajor A. (1988) Sections of the unit ball of ℓ n q . J. Funct. Anal. 80:109–123
Milman V.D., Pajor A. (1989) Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Lindenstrauss J., Milman V.D. (Eds.) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, 1376, Springer, Heidelberg, 64–104
Polya G. (1913) Berechnung eines bestimmten integrals. Math. Ann. 74:204–212
Romik D. Randomized central limit theorems. Preprint
Sudakov V.N. (1978) Typical distributions of linear functionals in finite-dimensional spaces of higher dimension. Soviet Math. Dokl. 19:1578–1582
Vaaler J.D. (1979) A geometric inequality with applications to linear forms. Pacific J. Math. 83:543–553
Voigt J. (2000) A concentration of mass property for isotropic convex bodies in high dimensions. Israel J. Math. 115:235–251
von Weizsäcker H. (1997) Sudakov's typical marginals, random linear functionals and a conditional central limit theorem. Prob. Theory and Related Fields 107:313–324
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2000 Springer-Verlag
About this chapter
Cite this chapter
Koldobsky, A., Lifshits, M. (2000). Average volume of sections of star bodies. In: Milman, V.D., Schechtman, G. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1745. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0107212
Download citation
DOI: https://doi.org/10.1007/BFb0107212
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41070-6
Online ISBN: 978-3-540-45392-5
eBook Packages: Springer Book Archive