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

TL;DR

This paper derives explicit particle-size distribution formulas for convex bodies in three-dimensional space, generalizing Wicksell's problem, using integral equations and illustrating with examples.

Contribution

It introduces explicit formulas for size distributions of convex bodies in 3D, extending classical problems with new integral equation solutions.

Findings

Explicit distribution formulas derived for convex bodies in 3D

Application of the Method of Model Solutions to Santaló's integral equations

Partial results on existence and uniqueness of solutions

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.