Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows


In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.

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Aramyan, R. (2015) Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows. American Journal of Computational Mathematics, 5, 86-95. doi: 10.4236/ajcm.2015.52007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Minkowski, H. (1911) Theorie der konvexen Korper, insbesondere Begrundung ihresb Oberflachenbergriffs. Ges. Abh., 2, Leipzig, Teubner, 131-229.
[2] Blaschke, W. (1923) Vorlesungen uber Differentialgeometrie. II. Affine Differentialgeometrie, Springer-Verlag, Berlin.
[3] Pogorelov, A.V. (1969) Exterior Geometry of Convex Surfaces [in Russian]. Nauka, Moscow.
[4] Alexandrov, A.D. (1956) Uniqueness Theorems for Surfaces in the Large [in Russian]. Vesti Leningrad State University, 19, 25-40.
[5] Bakelman, I.Ya., Verner, A.L. and Kantor, B.E. (1973) Differential Geometry in the Large [in Russian]. Nauka, Moskow.
[6] Firey, W.J. (1970) Intermediate Christoffel-Minkowski Problems for Figures of Revolution. Israel Journal of Mathematics, 8, 384-390.
[7] Berg, C. (1969) Corps convexes et potentiels spheriques. Matematisk-fysiske Meddelelser Udgivet af. Det Kongelige Danske Videnskabernes Selska, 37, 64.
[8] Wiel, W. and Schneider, R. (1983) Zonoids and Related Topics. In: Gruber, P. and Wills, J., Eds., Convexity and Its Applications, Birkhauser, Basel, 296-317.
[9] Gardner R.J. and Milanfar, P. (2003) Reconstruction of Convex Bodies from Brightness Functions. Discrete & Computational Geometry, 29, 279-303.
[10] Ryabogin, D. and Zvavich, A. (2004) Reconstruction of Convex Bodies of Revolution from the Areas of Their Shadows. Archiv der Mathematik, 5, 450-460.
[11] Leichtweiz, K. (1980) Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin.
[12] Ambartzumian, R.V. (1990) Factorization Calculus and Geometrical Probability. Cambridge University Press, Cambridge.
[13] Aramyan, R.H. (2001) An Approach to Generalized Funk Equations I [in Russian]. Izvestiya Akademii Nauk Armenii. Matematika [English Translation: Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 36, 47-58.
[14] Aramyan, R.H. (2010) Generalized Radon Transform on the Sphere. Analysis International Mathematical Journal of Analysis and Its Applications, 30, 271-284.
[15] Aramyan, R.H. (2010) Solution of an Integral Equation by Consistency Method. Lithuanian Mathematical Journal, 50, 133-139.
[16] Blaschke, W. (1956) Kreis und Kugel, (Veit, Leipzig). 2nd Edition, De Gruyter, Berlin.

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