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Centrally symmetric convex bodies and Radon transforms on higher order Grassmannians

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Paul Goodey  and
Wolfgang Weil
Paul Goodey
Affiliation:
Professor P. Goodey, Mathematics Department, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
Wolfgang Weil
Affiliation:
Professor W. Weil, Mathematisches Institut II, Universität Karlsruhe, Englerstrasse 2, 7500 Karlsruhe I, Germany.
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Extract

The purpose of this work is to investigate the relationship between Radon transforms and centrally symmetric convex bodies. Because of the injectivity properties of the Radon transform it is natural to consider transforms on the sphere separately from those on the higher order Grassmannians. Here we shall concentrate on the latter, whilst the former will be the subject of another article presently in preparation, Goodey and Weil [1991].

MSC classification

Secondary: 52A40: Inequalities and extremum problems

Information

Type
Research Article
Information
Mathematika , Volume 38 , Issue 1 , June 1991 , pp. 117 - 133
Copyright
Copyright © University College London 1991

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References

Aleksandrov, A. D.. Zur Theorie der gemischten Volumina von konvexen Körpern I, II, III (Russian with German summary). Mat. Sbornik N. S., 2 (1937), 947–972, 1205-1238, N. S., 3 (1938), 27-46.Google Scholar
Betke, U. and Goodey, P. R.. Continuous translation invariant valuations on convex bodies. Abh. Math. Sem. Hamburg, 54 (1984), 95–105.CrossRefGoogle Scholar
Bonnesen, T. and Fenchel, W.. Theorie der konvexen Korper (Springer, Berlin, 1934).Google Scholar
Fenchel, W. and Jessen, B.. Mengenfunktionen und konvexe Körper. Danske Vid. Selsk. Mat. Fys. Medd., 16 (1938), 1–31.Google Scholar
Gelfand, I. M., Graev, M. I. and Rosu, R.. The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. Jour. Operator Theory, 12 (1984), 359–383.Google Scholar
Goodey, P. R.. Centrally symmetric convex sets and mixed volumes. Mathematika, 24 (1977), 193–198.CrossRefGoogle Scholar
Goodey, P. R.. Centrally symmetric convex bodies. Mathematika, 31 (1984), 305–322.CrossRefGoogle Scholar
Goodey, P. R. and Howard, R.. Processes of flats induced by higher dimensional processes. Advances in Mathematics, 80 (1990), 92–109.CrossRefGoogle Scholar
Goodey, P. R. and Weil, W.. Translative integral formulae for convex bodies. Aequationes Math., 34 (1987), 64–77.CrossRefGoogle Scholar
Goodey, P. R. and Weil, W.. Centrally symmetric convex bodies and the spherical Radon transform. In preparation.Google Scholar
Grinberg, E.. Radon transforms on higher rank Grassmannians. Jour. Differential Geometry, 24 (1986), 53–68.Google Scholar
Helgason, S.. The Radon Transform (Birkhauser, Boston, 1980).CrossRefGoogle Scholar
Helgason, S.. Groups and Geometric Analysis (Academic Press, Orlando etc., 1984).Google Scholar
Leichtweiss, K.. Konvexe Mengen (Springer, Berlin, 1980).CrossRefGoogle Scholar
Matheron, G.. Un theoreme d'unicite pour les hyperplans poissoniens. J. Appl. Prob., 11 (1974), 184–189.CrossRefGoogle Scholar
Matheron, G.. Random Sets and Integral Geometry (Wiley, New York etc., 1975).Google Scholar
McMullen, P.. Polytopes with centrally symmetric faces. Israel Jour. Math., 8 (1970), 194–196.CrossRefGoogle Scholar
Rham, G. de. Differentiate Manifolds (Springer, New York, 1984).CrossRefGoogle Scholar
Shephard, G. C.. Approximation problems for convex polyhedra. Mathematika, 11 (1964), 9–18.CrossRefGoogle Scholar
Schneider, R.. Zu einem Problem von Shephard iiber die Projektionen konvexer Korper. Math. Zeit., 101 (1967), 71–82.CrossRefGoogle Scholar
Schneider, R.. Boundary structure and curvature of convex bodies. Contributions to Geometry (eds. Tolke, J. and Wills, J. M.), pp. 13–59 (Birkhauser, Basel, 1979).CrossRefGoogle Scholar
Schneider, R. and Weil, W.. Zonoids and related topics. Convexity and its Applications (eds. Gruber, P. and Wills, J. M.), pp. 296–317 (Birkhauser, Basel, 1983).CrossRefGoogle Scholar
Weil, W.. Centrally symmetric convex bodies and distributions. Israel Jour. Math., 24 (1976), 352–367.CrossRefGoogle Scholar
Weil, W.. Centrally symmetric convex bodies and distributions II. Israel Jour. Math., 32 (1979), 173–182.CrossRefGoogle Scholar
Weil, W.. Zonoide und verwandte Klassen konvexer Korper. Monatsh. Math., 94 (1982), 73–84.CrossRefGoogle Scholar
Weil, W.. Iterations of translative integral formulae and non-isotropic Poisson processes of particles. Math. Zeit., 205 (1990), 531–549.CrossRefGoogle Scholar