Determination of convex bodies by values of parallel X-ray of order i or parallel i-chord functions.

Geometric Tomography as we know it today began in 1963 with the following Hammers question: Suppose there is a convex hole in an otherwise homogeneous solid and that X-ray pictures taken 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 finite point source? b. The X-rays are assumed parallel? The point X-ray was generalized later by defining point X-ray of order i for any real number i, but the parallel version was never defined. Situation with the i-chord functions is quite similar. Therefore, at the beginning of the talk we will give the definitions of parallel X-ray of order i and the parallel i-chord functions. Later on, basic properties and results concerning determination of convex bodies by parallel X-ray of order i will be presented