On Particle-Size Distribution of Convex Similar Bodies in $${\mathbb {R}}^3$$ R 3

We have solved an old problem posed by Santaló of determining the size distribution of particles derived from the size distribution of their sections. We give an explicit form of particle-size distributions of convex similar bodies for random planes and random lines, which naturally generalize famous Wicksell’s corpuscle problem. The results are achieved by applying the method of model solutions for solving well-known Santaló’s integral equations. We give a partial result related to the question of the existence and uniqueness of these solutions. We also emphasize that the original form of solution of Wicksell’s problem is insufficient. We finally illustrate our approach in several examples.