Wicksell's corpuscle problem in spaces of constant curvature

Abstract

In this paper, we study Wicksell's corpuscle problem in spaces of constant curvature, thus extending the classical Euclidean framework. We consider a particle process of balls with random radii in such a space, assumed to be invariant under the action of the full isometry group. We refer to the process of intersections of these balls with a fixed totally geodesic hypersurface as the induced process. We first derive a section formula expressing the distribution of the induced process in terms of that of the original process. Conversely, assuming that the distribution of the induced process is known, we establish an inversion formula that recovers the distribution of the original process. Finally, we show that in the limit as the curvature of the space tends to zero, our section and inversion formulas converge to the classical Euclidean results.