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Stability and Separation in Volume ComparisonProblems

Published online by Cambridge University Press:  28 January 2013

A. Koldobsky
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Abstract

We review recent stability and separation results in volume comparison problems and usethem to prove several hyperplane inequalities for intersection and projection bodies.

Keywords

convex bodiessectionsprojections
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 156 - 169
Copyright
© EDP Sciences, 2013

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References

K. Ball. Some remarks on the geometry of convex sets. Geometric aspects of functional analysis (1986/87), Lecture Notes in Math. 1317, Springer-Verlag, Berlin-Heidelberg-New York, 1988, 224–231.
Ball, K.. Shadows of convex bodies. Trans. Amer. Math. Soc. 327 (1991), 891–901. CrossRefGoogle Scholar
Ball, K.. Logarithmically concave functions and sections of convex sets in R n. Studia Math. 88 (1988), 69–84. Google Scholar
Bourgain, J.. On high-dimensional maximal functions associated to convex bodies. Amer. J. Math. 108 (1986), 1467–1476. CrossRefGoogle Scholar
J.Bourgain. Geometry of Banach spaces and harmonic analysis. Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 871–878.
J.Bourgain. On the distribution of polynomials on high–dimensional convex sets. Geometric aspects of functional analysis, Israel seminar (198990), Lecture Notes in Math. 1469 Springer, Berlin, 1991, 127–137.
Bourgain, J.. On the Busemann-Petty problem for perturbations of the ball. Geom. Funct. Anal. 1 (1991), 1–13. CrossRefGoogle Scholar
Busemann, H., Petty, C. M.. Problems on convex bodies. Math. Scand. 4 (1956), 88–94. CrossRefGoogle Scholar
Gardner, R. J.. Intersection bodies and the Busemann-Petty problem. Trans. Amer. Math. Soc. 342 (1994), 435–445. CrossRefGoogle Scholar
Gardner, R. J.. A positive answer to the Busemann-Petty problem in three dimensions. Annals of Math. 140 (1994), 435–447. CrossRefGoogle Scholar
R. J. Gardner. Geometric tomography. Second edition, Cambridge University Press, Cambridge, 2006.
Gardner, R. J., Koldobsky, A., Schlumprecht, Th.. An analytic solution to the Busemann-Petty problem on sections of convex bodies. Annals of Math. 149 (1999), 691–703. CrossRefGoogle Scholar
I. M. Gelfand, G. E. Shilov. Generalized functions, vol. 1. Properties and operations. Academic Press, New York, 1964.
I. M. Gelfand, N. Ya. Vilenkin. Generalized functions, vol. 4. Applications of harmonic analysis. Academic Press, New York, 1964.
Giannopoulos, A.. A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies. Mathematika 37 (1990), 239–244. CrossRefGoogle Scholar
Goodey, P., Lutwak, E., Weil, W.. Functional analytic characterization of classes of convex bodies. Math. Z. 222 (1996), 363–381. CrossRefGoogle Scholar
Grinberg, E., Zhang, Gaoyong. Convolutions, transforms, and convex bodies. Proc. London Math. Soc. (3) 78 (1999), 77–115. CrossRefGoogle Scholar
Klartag, B.. On convex perturbations with a bounded isotropic con- stant. Geom. Funct. Anal. (GAFA) 16 (2006), 1274–1290. CrossRefGoogle Scholar
Koldobsky, A.. An application of the Fourier transform to sections of star bodies. Israel J. Math. 106 (1998), 157–164. CrossRefGoogle Scholar
Koldobsky, A.. Intersection bodies, positive definite distributions and the Busemann-Petty problem. Amer. J. Math. 120 (1998), 827–840. CrossRefGoogle Scholar
Koldobsky, A.. Intersection bodies in R4. Adv. Math. 136 (1998), 1–14. CrossRefGoogle Scholar
A. Koldobsky. Fourier analysis in convex geometry. Amer. Math. Soc., Providence RI, 2005.
Koldobsky, A.. A generalization of the Busemann-Petty problem on sections of convex bodies. Israel J. Math. 110 (1999), 75–91. CrossRefGoogle Scholar
Koldobsky, A.. Stability in the Busemann-Petty and Shephard problems. Adv. Math. 228 (2011), 2145–2161. CrossRefGoogle Scholar
Koldobsky, A.. Stability of volume comparison for complex convex bodies. Arch. Math. (Basel) 97 (2011), 91–98. CrossRefGoogle Scholar
Koldobsky, A.. A hyperplane inequality for measures of convex bodies in Rn,n ≤ 4. Dicrete Comput. Geom. 47 (2012), 538–547. CrossRefGoogle Scholar
A. Koldobsky, Dan Ma. Stability and slicing inequalities for intersection bodies. Geom. Dedicata, 2012, DOI : 10.1007/s10711-012-9729-x
A. Koldobsky, G. Paouris, M. Zymonopoulou. Complex intersection bodies. arXiv :1201.0437.
A. Koldobsky, M. Lifshits. Average volume of sections of star bodies. Geometric aspects of functional analysis, 119–146, Lecture Notes in Math., 1745, Springer, Berlin, 2000.
Koldobsky, A., Ryabogin, D., Zvavitch, A.. Projections of convex bodies and the Fourier transform. Israel J. Math. 139 (2004), 361–380. CrossRefGoogle Scholar
Koldobsky, A., Yaskin, V., Yaskina, M.. Modified Busemann-Petty problem on sections of convex bodies. Israel J. Math. 154 (2006), 191–207. CrossRefGoogle Scholar
A. Koldobsky, V. Yaskin. The interface between convex geometry and harmonic analysis. CBMS Regional Conference Series in Mathematics, 108, American Mathematical Society, Providence, RI, 2008.
Larman, D. G., Rogers, C. A.. The existence of a centrally symmetric convex body with central sections that are unexpectedly small. Mathematika 22 (1975), 164–175. CrossRefGoogle Scholar
Lutwak, E.. Intersection bodies and dual mixed volumes. Adv. Math. 71 (1988), 232–261. CrossRefGoogle Scholar
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 and V. Milman, Lecture Notes in Mathematics 1376, Springer, Heidelberg, 1989, pp. 64–104.
Papadimitrakis, M.. On the Busemann-Petty problem about convex, centrally symmetric bodies in Rn. Mathematika 39 (1992), 258–266. CrossRefGoogle Scholar
C. M. Petty. Projection bodies. Proc. Coll. Convexity (Copenhagen 1965), Kobenhavns Univ. Mat. Inst., 234-241.
Schneider, R.. Zu einem problem von Shephard über die projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. CrossRefGoogle Scholar
R. Schneider. Convex bodies : the Brunn-Minkowski theory. Cambridge University Press, Cambridge, 1993.
Shephard, G. C.. Shadow systems of convex bodies. Israel J. Math. 2 (1964), 229-306. CrossRefGoogle Scholar
Yaskin, V.. Modified Shephard’s problem on projections of convex bodies. Israel J. Math. 168 (2008), 221–238. CrossRefGoogle Scholar
Zhang, Gaoyong. Centered bodies and dual mixed volumes. Trans. Amer. Math. Soc. 345 (1994), 777–801. CrossRefGoogle Scholar
Zhang, Gaoyong. Intersection bodies and Busemann-Petty inequalities in R4. Annals of Math. 140 (1994), 331–346. CrossRefGoogle Scholar
Zhang, Gaoyong. A positive answer to the Busemann-Petty problem in four dimensions. Annals of Math. 149 (1999), 535–543. CrossRefGoogle Scholar
Zhang, Gaoyong. Sections of convex bodies. Amer. J. Math. 118 (1996), 319–340. CrossRefGoogle Scholar
Zvavitch, A.. The Busemann-Petty problem for arbitrary measures. Math. Ann. 331 (2005), 867–887. CrossRefGoogle Scholar