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.

我们已经解决了Santaló提出的一个老问题，即确定由其截面尺寸分布得出的颗粒尺寸分布。对于随机平面和随机线，我们给出了凸形相似物体的粒度分布的显式形式，这自然地推广了著名的Wicksell的小球问题。通过应用模型解决方案的方法来解决著名的Santaló积分方程，可以达到上述结果。我们给出与这些解决方案的存在性和唯一性有关的部分结果。我们还强调，维克塞尔问题的原始解决方案形式不足。最后，我们通过几个示例来说明我们的方法。