Abstract
A \(\sqrt{n}\) estimate in the hyperplane problem with arbitrary measures has recently been proved in [12]. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these inequalities from stability in comparison problems for different generalizations of intersection bodies.
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References
K. Ball, Logarithmically concave functions and sections of convex sets in \(R^{n}\). Studia Math. 88, 69–84 (1988)
J. Bourgain, On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108, 1467–1476 (1986)
J. Bourgain, Geometry of Banach spaces and harmonic analysis, in Proceedings of the International Congress of Mathematicians (Berkeley, 1986) (American Mathematical Society, Providence, 1987), pp. 871–878
J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, in Geometric Aspects of Functional Analysis, Israel Seminar (1989–90). Lecture Notes in Mathematics, vol. 1469 (Springer, Berlin, 1991), pp. 127–137
S. Brazitikos, A. Giannopoulos, P. Valettas, B. Vritsiou, Geometry of Isotropic Log-Concave Measures (American Mathematical Society, Providence, 2014)
R.J. Gardner, Geometric Tomography, 2nd edn. (Cambridge University Press, Cambridge, 2006)
E. Grinberg, G. Zhang, Convolutions, transforms and convex bodies. Proc. Lond. Math. Soc. 78, 77–115 (1999)
F. John, Extremum Problems with Inequalities as Subsidiary Conditions. Courant Anniversary Volume (Interscience, New York, 1948), pp. 187–204
B. Klartag, On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16, 1274–1290 (2006)
A. Koldobsky, Fourier Analysis in Convex Geometry (American Mathematical Society, Providence, 2005)
A. Koldobsky, A hyperplane inequality for measures of convex bodies in \(\mathbb{R}^{n},n \leq 4\). Dicrete Comput. Geom. 47, 538–547 (2012)
A. Koldobsky, A \(\sqrt{n}\) estimate for measures of hyperplane sections of convex bodies. Adv. Math. 254, 33–40 (2014)
A. Koldobsky, D. Ma, Stability and slicing inequalities for intersection bodies. Geom. Dedicata 162, 325–335 (2013)
A. Koldobsky, G. Paouris, M. Zymonopoulou, Complex intersection bodies. J. Lond. Math. Soc. 88(2), 538–562 (2013)
E. Lutwak, Intersection bodies and dual mixed volumes. Adv. Math. 71, 232–261 (1988)
E. Milman, Generalized intersection bodies. J. Funct. Anal. 240(2), 530–567 (2006)
V. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in Geometric Aspects of Functional Analysis, ed. by J. Lindenstrauss, V. Milman. Lecture Notes in Mathematics, vol. 1376 (Springer, Heidelberg, 1989), pp. 64–104
G. Zhang, Section of convex bodies. Am. J. Math. 118, 319–340 (1996)
Acknowledgements
I wish to thank the US National Science Foundation for support through grant DMS-1265155.
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Koldobsky, A. (2014). Estimates for Measures of Sections of Convex Bodies. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_17
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DOI: https://doi.org/10.1007/978-3-319-09477-9_17
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