On particle-size distribution of convex similar bodies in $\mathbb{R}^3$

Abstract: We give an explicit form of particle-size distributions of convex similar bodies for random plane and random line, which naturally generalize famous Wicksell's corpuscle problem. The results are achieved by applying the Method of Model Solutions for solving well-known Santal\'o's integral equations. We also give a partial result related to the question of existence and uniqueness of these solutions. We finally illustrate our approach on several examples.