Volčič, Aljoša, Kolokvij maj 2016

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Hammer's X-ray problem

Aljoša Volčič

University of Calabria, Italy

Suppose there is a convex hole in an otherwise homogeneous solid and that X-ray pictures are so sharp that the ``darkness" at each point determines the length of a chord 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 to be 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.

These questions had to wait the first half of the eighties to get the first answers. All the results quoted below refer to the plane.

Theorem (Gardner-McMullen, 1980) There exist four directions such that parallel X-rays taken in that directions determine all planar convex bodies.

Theorem (Falconer/Gardner, 1983) Given two points p and q and a convex body K whose interior intersects the line pq, K is uniquely determined by its point X-rays at p and q if

i) p, q \in \mbox{int}K;

ii) p,q \not \in K and the component of pq \setminus \{p,q\} is specified.

Theorem (Volčič, 1986) If p,q,r are three non-collinear points and K is a convex body which does not contain them, then K is uniquely determined by its point X-rays taken at p,q,r.

I will present several of the most interesting open problems and some recent developements of the theory as the Barker-Larman problem and two results concerning higher dimensions.

Osebna orodja