Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T10:25:59.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability estimates for star bodies in terms of their intersection bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

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

Abstracat

The paper deals with the problem of estimating the distance, in radial or Hausdorff metrics, between two centred star bodies of Rd, d≤3, in terms of the distance between the corresponding intersection bodies.

MSC classification

Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 287 - 303
Copyright
Copyright © University College London 1998

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

A.Aubin, T.. Nonlinear analysis on manifolds. Monge–Ampére equations (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar
BL.Bourgain, J. and Lindenstrauss, J.. Projection bodies. Geometric Aspects of Functional Analysis (Lindenstrauss, J., Milman, V. D., eds.), Lecture Notes in Math. 1317 (Springer, Berlin, 1988), 250–270.CrossRefGoogle Scholar
CNS.Caffarelli, L., Nirenberg, L. and Spruck, J.. Nonlinear second order elliptic equations IV. Starshaped compact Weingarten hypersurfaces. Current topics in partial differential equations (Ohya, Y., Kasahara, K., Shimakura, N., eds.), (Kinokunize, Tokyo, 1986), 1–26.Google Scholar
Cl.Campi, S.. On the reconstruction of a star-shaped body from its “half-volumes”. J. Austral. Math. Soc. (A), 37 (1984), 243257.CrossRefGoogle Scholar
C2.Campi, S.. Recovering a centred convex body from the areas of its shadows: a stability estimate. Ann. Mat. Pura Appl., 151 (1988), 289302.CrossRefGoogle Scholar
CW.Coifman, R. R. and Weiss, G. L.. Representations of compact groups and spherical harmonics. Enseign. Math., 2 (1968), 121–173.Google Scholar
Gl.Gardner, R. J.. Geometric Tomography (Cambridge University Press, Cambridge, 1995).Google Scholar
G2.Gardner, R. J.. Intersection bodies and the Busemann Petty problem. Trans. Amer. Math. Soc, 342 (1994), 435–445.CrossRefGoogle Scholar
G3.Gardner, R. J.. On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull. Amer. Math. Soc, 30 (1994), 222226.CrossRefGoogle Scholar
G4.Gardner, R. J.. A positive answer to the Busemann–Petty problem in three dimensions. Ann. of Math., 140 (1994), 435–447.CrossRefGoogle Scholar
GG.Goodey, P. R. and 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.CrossRefGoogle Scholar
GW1.Goodey, P. and Weil, W.. Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom., 35 (1992), 675–688.CrossRefGoogle Scholar
GW2.Goodey, P. and Weil, W.. Intersection bodies and ellipsoids. Mathematika, 42 (1995), 295–304.CrossRefGoogle Scholar
GR.Gradshteyn, I. S. and Rizhik, I. M.. Table of integrals, series and products (Academic Press, New York, 1980).Google Scholar
Grw.Greenwald, H. C.. Lipschitz spaces on the surface of the unit sphere in Euclidean n-space. Pacific J. Math., 50 (1974), 6380.CrossRefGoogle Scholar
Grm.Groemer, H.. Stability results for convex bodies and related spherical integral transformations. Adv. Math., 109 (1994), 45–74.CrossRefGoogle Scholar
Kl.Koldobsky, A.. Intersection bodies in R4. Adv. Math., 136 (1998), 114.CrossRefGoogle Scholar
K2.Koldobsky, A.. Intersection bodies, positive definite distributions and the Busemann Petty problem. Amer. J. Math., 120 (1998), 827840.CrossRefGoogle Scholar
Lo.Lorentz, G. G., Bernstein polynomials (University of Toronto Press, Toronto, 1953).Google Scholar
Lu.Lutwak, E.. Intersection bodies and dual mixed volumes. Adv. Math., 71 (1988), 232261.CrossRefGoogle Scholar
O.Oliker, V. I.. Hypersurfaces in Rn+1 with prescribed Gaussian curvature and related equations of Monge–Ampere type. Comm. Partial Differential Equations, 9 (8) (1984), 807–838.CrossRefGoogle Scholar
s1.Schneider, R.. Functions on a sphere with vanishing integrals over certain subspheres. J. Math. Anal. Appl., 26 (1969), 381–384.CrossRefGoogle Scholar
52.Schneider, R.. Functional equations connected with rotations and their applications. Enseign. Math. (2), 6 (1970), 297305.Google Scholar
St.Strichartz, R.. If estimates for Radon transforms in Euclidean and non-Euclidean spaces. Duke Math. J., 48 (1981), 699–727.CrossRefGoogle Scholar
TW.Treibergs, A. E. and Wei, S. W.. Embedded hyperspheres with prescribed mean curvature. J. Differential Geom., 18 (1983), 513–521.CrossRefGoogle Scholar
U.Urbas, J. I.. On the expansions of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z., 205 (1990), 355–372.CrossRefGoogle Scholar
VI.Vilenkin, N. Ja.. Spherical functions and the theory of group representations (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
Vt.Vitale, R.. Lp metrics for compact convex sets. J. Approx. Theory, 45 (1985), 280–287.CrossRefGoogle Scholar
Zl.Zhang, G.. Centered bodies and dual mixed volumes. Trans. Amer. Math. Soc, 345 (1994), 777–801.CrossRefGoogle Scholar
Z2.Zhang, G.. Intersection bodies and the Busemann–Petty inequalities in R4. Ann. of Math., 140 (1994), 31–46.CrossRefGoogle Scholar
Z3.Zhang, G.. A positive answer to the Busemann–Petty problem in four dimensions. Mathematika. (to appear).Google Scholar