Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T14:20:07.319Z Has data issue: false hasContentIssue false
  • English
  • Français

The Lower Dimensional Busemann-Petty Problem with Weights

Part of: Integral transforms, operational calculus General convexity

Published online by Cambridge University Press:  21 December 2009

Boris Rubin
Boris Rubin
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, U.S.A. E-mail: borisr@math.lsu.edu
Get access

Abstract

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.

MSC classification

Secondary: 52A38: Length, area, volume 44A12: Radon transform
Type
Research Article
Information
Mathematika , Volume 53 , Issue 2 , December 2006 , pp. 235 - 245
Copyright
Copyright © University College London 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bourgain, J. and Zhang, G., On a generalization of the Busemann-Petty problem. In Convex Geometric Analysis (Berkeley, CA, 1996), Math. Sci. Res. Inst. Publ. 34, Cambridge University Press (Cambridge, 1999), 65–76.Google Scholar
2Busemann, H. and Petty, C. M., Problems on convex bodies. Math. Scand. 4 (1956), 88–94.CrossRefGoogle Scholar
3Gardner, R. J., Geometric Tomography. Cambridge University Press (New York, 1995) (second edition, 2006).Google Scholar
4Gardner, R. J., Koldobsky, A. and Schlumprecht, T., An analytic solution to the Busemann-Petty problem on sections of convex bodies. Annals Math. 149 (1999), 691–703.CrossRefGoogle Scholar
5Goodey, P. R., Lutwak, E. and Weil, W., Functional analytic characterizations of classes of convex bodies. Math. Z. 222 (1996), 363–381.CrossRefGoogle Scholar
6Grinberg, E. L. and Zhang, G., Convolutions, transforms, and convex bodies. Proc. London Math. Soc. (3) 78 (1999), 77–115.CrossRefGoogle Scholar
7Helgason, S., The Radon Transform (2nd ed.). Birkhäuser (Boston, 1999).CrossRefGoogle Scholar
8Koldobsky, A., A generalization of the Busemann-Petty problem on sections of convex bodies. Israel J. Math. 110 (1999), 75–91.CrossRefGoogle Scholar
9Koldobsky, A., A functional analytic approach to intersection bodies. Geom. Funct. Anal. 10 (2000), 1507–1526.CrossRefGoogle Scholar
10Koldobsky, A., Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs 116, Amer. Math. Soc. (2005).Google Scholar
11Lutwak, E., Intersection bodies and dual mixed volumes. Advances Math. 71 (1988), 232–261.CrossRefGoogle Scholar
12Rubin, B., Inversion formulas for the spherical Radon transform and the generalized cosine transform. Advances Appl. Math. 29 (2002), 471–497.CrossRefGoogle Scholar
13Rubin, B., Notes on Radon transforms in integral geometry. Fractional Calculus Appl. Analysis 6 (2003), 25–72.Google Scholar
14Rubin, B. and Zhang, G., Generalizations of the Busemann-Petty problem for sections of convex bodies. J. Funct. Anal. 213 (2004), 473–501.CrossRefGoogle Scholar
15Yaskin, V., The Busemann-Petty problem in hyperbolic and spherical spaces. Advances Math. 203 (2006), 537–553.CrossRefGoogle Scholar
16Yaskin, V., A solution to the lower dimensional Busemann-Petty problem in the hyperbolic space. J. Geom. Anal. 16 (2006), 735–745.CrossRefGoogle Scholar
17Zhang, G., Sections of convex bodies. Amer. J. Math. 118 (1996), 319–340.CrossRefGoogle Scholar
18Zhang, G., A positive solution to the Busemann-Petty problem in ℝ4. Annals Math. (2) 149 (1999), 535–543.CrossRefGoogle Scholar
19Zvavitch, A., Gaussian measure of sections of convex bodies. Advances Math. 188 (2004), 124–136.CrossRefGoogle Scholar
20Zvavitch, A., The Busemann-Petty problem for arbitrary measures. Math. Ann. 331 (2005), 867–887.CrossRefGoogle Scholar
21Milman, E., Generalized intersection bodies. J. Funct. Anal. 240 (2006), 530–567.CrossRefGoogle Scholar
22Milman, E., Generalized intersection bodies are not equivalent (2007) math.FA/0701779.Google Scholar
23Rubin, B., Intersection bodies and generalized cosine transforms (2007) arXiv:0704.006 1.Google Scholar