Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-01T00:03:17.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Convex intersection bodies in three and four dimensions

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Stefano Campi
Stefano Campi
Affiliation:
Departimento di Matematica Pura e Applicata “G. Vitali”, Università degli Studi di Modena, Via Campi 213/B, 41100 Modena. Italy.
Get access

Abstract

The paper shows that no origin-symmetric convex polyhedron in R3 is the intersection body of a star body. It is shown also that every originsymmetric convex body in Rd, for d = 3 and 4, can be seen as the intersection body of a star-shaped set whose radial function satisfies conditions related to suitable non-integer Sobolev classes.

MSC classification

Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 15 - 27
Copyright
Copyright © University College London 1999

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

[B]Ball, K.. Cube slicing in Rn. Proc. Amer. Math. Soc., 97 (1986), 465–473.Google Scholar
[Bo]Bourgain, J.. On the Busemann Petty problem for perturbations of the ball. Geom. Functional Anal., 1 (1991), 1–13.CrossRefGoogle Scholar
[C]Campi, S.. Stability estimates for star bodies in terms of their intersection bodies. Mathematika, 45 (1988), 287–303.CrossRefGoogle Scholar
[FGW]Fallert, H., Goodey, P. and Weil, W.. Spherical projections and centrally symmetric sets. Adv. Math., 129 (1997), 301–322.CrossRefGoogle Scholar
[G1]Gardner, R. J.. Intersection bodies and the Busemann Petty problem. Trans. Amer. Math. Soc., 342 (1994), 435–445.CrossRefGoogle Scholar
[G2]Gardner, R. J.. A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math., 140 (1994), 435–447.CrossRefGoogle Scholar
[G3]Gardner, R. J.. Geometric Tomography. (Cambridge University Press, 1995).Google Scholar
[Gi]Giannopoulos, A. A.. A note on a problem by H. Busemann and C. M. Petty concerning sections of symmetric convex bodies. Mathematika, 37 (1990), 239–244.Google Scholar
[GP]Giannopoulos, A. A. and Papadimitrakis, M.. The Busemann-Petty problem. Preprint.Google Scholar
[GG]Goodey, P., Groemer, H.. Stability results for first order projection bodies. Proc. Amer. Math. Soc., 109 (1990), 1103–1114.CrossRefGoogle Scholar
[GLW]Goodey, P., Lutwak, E. and Weil, W.. Functional analytic characterizations of classes of convex bodies. Math. Z., 222 (1996), 363–381.Google Scholar
[GW]Goodey, P. and Weil, W.,. Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom., 35 (1992), 675–688.Google Scholar
[Gr]Groemer, H.. Geometric Applications of Fourier Series and Spherical Harmonics. (Cambridge University Press, 1996).CrossRefGoogle Scholar
[H]Helgason, S.. Groups and Geometric Analysis. (Academic Press, Orlando, 1984).Google Scholar
[Kl]Klain, D. A.,. Star valuations and dual mixed volumes. Adv. Math., 121 (1996), 80–101.Google Scholar
[K1]Koldobsky, A.,. Intersection bodies in R 4. Adv. Math., 136 (1998), 1–14.CrossRefGoogle Scholar
[K2]Koldobsky, A.,. Intersection bodies, positive definite distributions and the Busemann-Petty problem. Amer. J. Math., 120 (1998), 827–840.Google Scholar
[LR]Larman, D. G. and Rogers, C. A.,. The existence of a centrally symmetric convex body with central sections that are unexpectedly small. Mathematika, 22 (1975), 164–175.Google Scholar
[L]Lutwak, E.,. Intersection bodies and dual mixed volumes. Adv. Math., 71 (1988), 232–261.CrossRefGoogle Scholar
[P]Papadimitrakis, N.,. On the Busemann-Petty problem about convex, centrally symmetric bodies in Rn. Mathematika, 39 (1992), 258–266.Google Scholar
[S1]Schneider, R.,. Functions on a sphere with vanishing integrals over certain subspheres. J. Math. Anal. Appl., 26 (1969), 381–384.CrossRefGoogle Scholar
[S2]Schneider, R.,. Functional equations connected with rotations and their applications. Einsegn. Math., (2) 6(1970), 297–305.Google Scholar
[St]Strichartz, R.,. Lp-estimates for Radon transforms in Euclidean and non-Euclidean spaces. Duke Math. J., 48(1981), 699–727.Google Scholar
[T]Triebel, H.. Theory of Function Spaces, Vol II. (Birkaiiser Verlag, Basel-Boston-Berlin, 1992).CrossRefGoogle Scholar
[Z1]Zhang, G.,. Intersection bodies and the Busemann-Petty inequalities in R 4. Ann. Math., 140(1994), 31–46.CrossRefGoogle Scholar
[Z2]Zhang, G.,. Centered bodies and dual mixed volumes. Trans. Amer. Math. Soc., 345(1994), 777–801.Google Scholar
[Z3]Zhang, G.,. A positive answer to the Busemann-Petty problem in R 4. Ann. Math., 149(1999), 535–543.CrossRefGoogle Scholar
[Z4]Zhang, G.,. Intersection bodies and polytopes. Mathematika (this volume), 29–34.CrossRefGoogle Scholar