Geometric Tomography of Convex Cones

The parallel X-ray of a convex set K⊂ℝ ⁿ in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex polyhedral cones in ℝ³ are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal.