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Uniqueness theorems for convex bodies in non-Euclidean spaces

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Paolo Dulio  and
Carla Peri
Paolo Dulio
Affiliation:
Dipartimento di Matematica “F. Brioschi”, Politeenico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano E-mail:paodul@mate.polimi.it
Carla Peri
Affiliation:
Università Cattolica S.C., Largo Gemelli 1, I-20123 Milano E-mail:carla.peri@unicatt.it
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Abstract

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

MSC classification

Secondary: 52A55: Spherical and hyperbolic convexity
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 13 - 31
Copyright
Copyright © University College London 2002

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References

1.Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-positive Curvature. Grundlehren der mathematischen Wissenschaften, vol. 319 (Springer, Heidelberg, 1999).CrossRefGoogle Scholar
2.Dulio, P. and Peri, C.. Invariant valuations on spherical star sets. Suppl. Rend. Mat. Palermo, Ser. II. n. 65. (2000). 81–92.Google Scholar
3.Falconer, K. J.. Hammer's X-ray problem and the stable manifold theorem. J. London Math. Soc. (2) 28 (1983), 149–160.CrossRefGoogle Scholar
4.Falconer, K. J.. X-ray problem for point sources. J. London Math. Soc. (3) 46 (1983), 241–262.Google Scholar
5.Fejes Tóth, G. and Kemnitz, A.. Characterization of centrally symmetric convex domains in planes of constant curvature. In Colloquia Mathematica Soc. János Bolyai, vol. 48, Intuitive Geometry (Siofok 1985), ed. by Böröczky, K. and Fejes Tóth, G. (North-Holland, Amsterdam. 1985). 179–189.Google Scholar
6.Gardner, R. J.. Chord functions of convex bodies. J. London Math. Soc. (2) 36 (1987), 314–326.CrossRefGoogle Scholar
7.Gardner, R. J.. Geometric Tomography (Cambridge University Press, New York, 1995). (For update go to http://www.ac.wwu.edu/gardner/)Google Scholar
8.Gardner, R. J. and Volčič, A.. Tomography of convex and star bodies. Advances Math., 108 (1994), 367–399.CrossRefGoogle Scholar
9.Helgason, S.. Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).Google Scholar
10.Kurusa, A.. Generalized X-ray pictures. Publ. Math. Debrecen 48 (1996), 193–198.CrossRefGoogle Scholar
11.Larman, D. G. and Tamvakis, N. K.. A characterization of centrally symmetric convex bodies in En. Geom. Dedicata, 10 (1981). 161–176.CrossRefGoogle Scholar
12.McMullen, P.. Konvexe Körpen Probleme. Oberwolfach, May, 1978.Google Scholar
13.Rogers, C. A.. An equichordal problem. Geom. Dedicata 10 (1981), 73–78.CrossRefGoogle Scholar
14.Volčič, A.. A three-point solution to Hammer's X-ray problem. J. London Math. Soc. (2) 34 (1986). 349–359.CrossRefGoogle Scholar
15.Willmore, T. J.. Riemannian Geometry (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar