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Suppose a number of X-ray
pictures is taken of the same object, but from different directions. One typically likes
to know to what degree the pictures determine the object and exactly when an object
is uniquely determined. Replacing picture taking by projections, that is, images
relative to specified mappings, these same problems are easily formulated for higher
dimensions and even for abstract spaces. The objects on hand might be data
structures.
With this general framework, starting from an arbitrary but fixed collection of
mappings, we study a new and very useful class of objects (sets) each of which is
uniquely determined by its projections. In the process, we disprove a previously
conjectured characterization of uniqueness relative to the one-dimensional projections
in Rn. For all situations where the underlying space is finite, a complete and rather
simple characterization of uniqueness is obtained.
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