Free Access
Math. Model. Nat. Phenom.
Volume 8, Number 1, 2013
Harmonic analysis
Page(s) 156 - 169
Published online 28 January 2013
  1. 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. [Google Scholar]
  2. K. Ball. Shadows of convex bodies. Trans. Amer. Math. Soc. 327 (1991), 891–901. [CrossRef] [MathSciNet] [Google Scholar]
  3. K. Ball. Logarithmically concave functions and sections of convex sets in Rn. Studia Math. 88 (1988), 69–84. [MathSciNet] [Google Scholar]
  4. J. Bourgain. On high-dimensional maximal functions associated to convex bodies. Amer. J. Math. 108 (1986), 1467–1476. [CrossRef] [MathSciNet] [Google Scholar]
  5. 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. [Google Scholar]
  6. 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. [Google Scholar]
  7. J. Bourgain. On the Busemann-Petty problem for perturbations of the ball. Geom. Funct. Anal. 1 (1991), 1–13. [CrossRef] [MathSciNet] [Google Scholar]
  8. H. Busemann, C. M. Petty. Problems on convex bodies. Math. Scand. 4 (1956), 88–94. [MathSciNet] [Google Scholar]
  9. R. J. Gardner. Intersection bodies and the Busemann-Petty problem. Trans. Amer. Math. Soc. 342 (1994), 435–445. [CrossRef] [MathSciNet] [Google Scholar]
  10. R. J. Gardner. A positive answer to the Busemann-Petty problem in three dimensions. Annals of Math. 140 (1994), 435–447. [CrossRef] [Google Scholar]
  11. R. J. Gardner. Geometric tomography. Second edition, Cambridge University Press, Cambridge, 2006. [Google Scholar]
  12. R. J. Gardner, A. Koldobsky, Th. Schlumprecht. An analytic solution to the Busemann-Petty problem on sections of convex bodies. Annals of Math. 149 (1999), 691–703. [CrossRef] [Google Scholar]
  13. I. M. Gelfand, G. E. Shilov. Generalized functions, vol. 1. Properties and operations. Academic Press, New York, 1964. [Google Scholar]
  14. I. M. Gelfand, N. Ya. Vilenkin. Generalized functions, vol. 4. Applications of harmonic analysis. Academic Press, New York, 1964. [Google Scholar]
  15. A. Giannopoulos. A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies. Mathematika 37 (1990), 239–244. [CrossRef] [MathSciNet] [Google Scholar]
  16. P. Goodey, E. Lutwak, W. Weil. Functional analytic characterization of classes of convex bodies. Math. Z. 222 (1996), 363–381. [CrossRef] [MathSciNet] [Google Scholar]
  17. E. Grinberg, Gaoyong Zhang. Convolutions, transforms, and convex bodies. Proc. London Math. Soc. (3) 78 (1999), 77–115. [CrossRef] [MathSciNet] [Google Scholar]
  18. B. Klartag. On convex perturbations with a bounded isotropic con- stant. Geom. Funct. Anal. (GAFA) 16 (2006), 1274–1290. [CrossRef] [Google Scholar]
  19. A. Koldobsky. An application of the Fourier transform to sections of star bodies. Israel J. Math. 106 (1998), 157–164. [CrossRef] [MathSciNet] [Google Scholar]
  20. A. Koldobsky. Intersection bodies, positive definite distributions and the Busemann-Petty problem. Amer. J. Math. 120 (1998), 827–840. [CrossRef] [MathSciNet] [Google Scholar]
  21. A. Koldobsky. Intersection bodies in R4. Adv. Math. 136 (1998), 1–14. [CrossRef] [MathSciNet] [Google Scholar]
  22. A. Koldobsky. Fourier analysis in convex geometry. Amer. Math. Soc., Providence RI, 2005. [Google Scholar]
  23. A. Koldobsky. A generalization of the Busemann-Petty problem on sections of convex bodies. Israel J. Math. 110 (1999), 75–91. [CrossRef] [MathSciNet] [Google Scholar]
  24. A. Koldobsky. Stability in the Busemann-Petty and Shephard problems. Adv. Math. 228 (2011), 2145–2161. [CrossRef] [MathSciNet] [Google Scholar]
  25. A. Koldobsky. Stability of volume comparison for complex convex bodies. Arch. Math. (Basel) 97 (2011), 91–98. [CrossRef] [MathSciNet] [Google Scholar]
  26. A. Koldobsky. A hyperplane inequality for measures of convex bodies in Rn,n ≤ 4. Dicrete Comput. Geom. 47 (2012), 538–547. [CrossRef] [Google Scholar]
  27. A. Koldobsky, Dan Ma. Stability and slicing inequalities for intersection bodies. Geom. Dedicata, 2012, DOI : 10.1007/s10711-012-9729-x [Google Scholar]
  28. A. Koldobsky, G. Paouris, M. Zymonopoulou. Complex intersection bodies. arXiv :1201.0437. [Google Scholar]
  29. 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. [Google Scholar]
  30. A. Koldobsky, D. Ryabogin, A. Zvavitch. Projections of convex bodies and the Fourier transform. Israel J. Math. 139 (2004), 361–380. [CrossRef] [MathSciNet] [Google Scholar]
  31. A. Koldobsky, V. Yaskin, M. Yaskina. Modified Busemann-Petty problem on sections of convex bodies. Israel J. Math. 154 (2006), 191–207. [CrossRef] [MathSciNet] [Google Scholar]
  32. 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. [Google Scholar]
  33. D. G. Larman, C. A. Rogers. The existence of a centrally symmetric convex body with central sections that are unexpectedly small. Mathematika 22 (1975), 164–175. [CrossRef] [MathSciNet] [Google Scholar]
  34. E. Lutwak. Intersection bodies and dual mixed volumes. Adv. Math. 71 (1988), 232–261. [CrossRef] [Google Scholar]
  35. 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. [Google Scholar]
  36. M. Papadimitrakis. On the Busemann-Petty problem about convex, centrally symmetric bodies in Rn. Mathematika 39 (1992), 258–266. [CrossRef] [MathSciNet] [Google Scholar]
  37. C. M. Petty. Projection bodies. Proc. Coll. Convexity (Copenhagen 1965), Kobenhavns Univ. Mat. Inst., 234-241. [Google Scholar]
  38. R. Schneider. Zu einem problem von Shephard über die projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. [CrossRef] [MathSciNet] [Google Scholar]
  39. R. Schneider. Convex bodies : the Brunn-Minkowski theory. Cambridge University Press, Cambridge, 1993. [Google Scholar]
  40. G. C. Shephard. Shadow systems of convex bodies. Israel J. Math. 2 (1964), 229-306. [CrossRef] [MathSciNet] [Google Scholar]
  41. V. Yaskin. Modified Shephard’s problem on projections of convex bodies. Israel J. Math. 168 (2008), 221–238. [CrossRef] [MathSciNet] [Google Scholar]
  42. Gaoyong Zhang. Centered bodies and dual mixed volumes. Trans. Amer. Math. Soc. 345 (1994), 777–801. [CrossRef] [MathSciNet] [Google Scholar]
  43. Gaoyong Zhang. Intersection bodies and Busemann-Petty inequalities in R4. Annals of Math. 140 (1994), 331–346. [CrossRef] [Google Scholar]
  44. Gaoyong Zhang. A positive answer to the Busemann-Petty problem in four dimensions. Annals of Math. 149 (1999), 535–543. [CrossRef] [Google Scholar]
  45. Gaoyong Zhang. Sections of convex bodies. Amer. J. Math. 118 (1996), 319–340. [CrossRef] [MathSciNet] [Google Scholar]
  46. A. Zvavitch. The Busemann-Petty problem for arbitrary measures. Math. Ann. 331 (2005), 867–887. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.