RADON TRANSFORM AND RECONSTRUCTION OF CONVEX BODIES

                  Aljo\v{s}a Vol\v{c}i\v{c}


Hammer posed in 1961, in a meeting on convexity
organized by AMS, the following problem:
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{\bf P.C. Hammer, ``Problem 2". Proc. Symp. Pure Math. VII: Convexity.
Amer. Math. Soc. 1963, pp. 498-499}

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{\sc The X-ray problems (Hammer)}. Suppose there is a convex hole $C$ in an
otherwise homogeneous solid and that X-ray pictures taken are so sharp
that the ``darkness"                                                     

at each point determines the lenght of a chord in $C$ along an X-ray line.
(No diffusion, please.) How many pictures must be taken to permit exact
reconstruction of the body if:

a. The X-rays issue from a finite point source?

b. The X-rays are assumed parallel?

For the planar counterpart, we have shown that two perpendicular directions
are insufficient for (b) and we conjecture that $3$ directions are
sufficient, although whether or not such directions must be strategically
chosen is also open.

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Note that the X-ray problem makes sense
also if the finite point is be taken in the interior
of $K$. One can imagine a device which rotates   
around the solid so that all the X-rays pass through
the given point $p\in \,$int$K$.

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This question stimulated  in the last twenty years
a vast literature on the problem itself and on some
variants.

This talk will make a brief history on the topic.