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Tomography is the study of the reconstruction of 2- and 3-dimensional objects from 1-dimensional slices. The Radon Transform is an important tool in tomography.
Rather surprisingly, there exist certain sets of four directions in Euclidean 
-space such that X-rays of a convex body
in these directions distinguish it from all other convex bodies.
See also Aleksandrov's Uniqueness Theorem, Brunn-Minkowski Inequality, Busemann-Petty Problem, Dvoretzky's Theorem, Radon Transform, Stereology
References
Gardner, R. J. ``Geometric Tomography.'' Not. Amer. Math. Soc. 42, 422-429, 1995.
Gardner, R. J. Geometric Tomography. New York: Cambridge University Press, 1995.
Tomography