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Intersection bodies and ellipsoids

Published online by Cambridge University Press:  26 February 2010

Paul Goodey
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, U.S.A.
Wolfgang Weil
Affiliation:
Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany
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Extract

In this paper we study various classes of centrally symmetric sets in d-dimensional Euclidean space Rd. As we will see, it is appropriate to focus our attention on those sets which have interior points.

MSC classification

Type
Research Article
Copyright
Copyright © University College London 1995

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References

Berg, Ch.. Corps convexes et potentiels sphériques. Mat.-Fys. Medd. Danske Vid. Selskab. 37, 6 (1969), 164.Google Scholar
Fallert, H., Goodey, P. and Weil, W.. Spherical projections and centrally symmetric sets. Advances in Math. To appear.Google Scholar
Gardner, R.. Intersection bodies and the Busemann-Petty problem. Trans. Amer. Math. Soc, 342 (1994), 435445.CrossRefGoogle Scholar
Gardner, R.. On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull. Amer. Math. Soc, 30 (1994b), 222226.CrossRefGoogle Scholar
Gardner, R.. A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math., 140 (1994), 435447.CrossRefGoogle Scholar
Gardner, R.. Geometric tomography, (Cambridge University Press, Cambridge, 1995).Google Scholar
Goodey, P., Lutwak, E. and Weil, W.. Functional analytic characterizations of classes of convex bodies. Math. Zeit. To appear.Google Scholar
Goodey, P. and Weil, W.. Distributions and valuations. Proc. London Math. Soc. (3), 49 (1984), 504516.CrossRefGoogle Scholar
Goodey, P. and Weil, W.. Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom. 35 (1992), 675688.CrossRefGoogle Scholar
Goodey, P. and Weil, W.. Zonoids and generalisations. In Handbook of Convex Geometry, edited by Gruber, P. and Wills, J. M. (Elsevier, Amsterdam 1993), 12971326.CrossRefGoogle Scholar
Helgason, S., The Radon transform (Birkh user, Basel, 1980).CrossRefGoogle Scholar
Lutwak, E.. Intersection bodies and dual mixed volumes. Adv. in Math., 71 (1988), 232261.CrossRefGoogle Scholar
Schneider, R.. Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc., 194 (1974), 5378.CrossRefGoogle Scholar
Schneider, R., Convex bodies: The Brunn-Minkowski theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
Schneider, R. and Weil, W.. Zonoids and related topics, In Convexity and Its Applications, edited by Gruber, P. and Wills, J. M., (Birkhauser, Basel, 1983), 296317.CrossRefGoogle Scholar
Schneider, R. and Wieacker, J. A.. Integral geometry in Minkowski spaces. Advances in Math. To appear.Google Scholar
Zhang, G.. Intersection bodies and the four-dimensional Busemann-Petty problem. Duke Math. J., 71 (1993), 233240.Google Scholar
Zhang, G.. Intersection bodies and Busemann-Petty inequalities in R4, Ann. Math., 140 (1994), 331346.CrossRefGoogle Scholar