Wicksell's corpuscle problem in spaces of constant curvature

TL;DR

This paper extends Wicksell's corpuscle problem to spaces of constant curvature, deriving formulas for the distribution of intersecting balls and their inverses, and shows convergence to classical Euclidean results as curvature approaches zero.

Contribution

It introduces a framework for analyzing Wicksell's problem in curved spaces, providing new section and inversion formulas that generalize Euclidean results.

Findings

Derived a section formula for the distribution of intersections in curved spaces.

Established an inversion formula to recover original distributions from intersections.

Proved convergence of formulas to Euclidean case as curvature tends to zero.

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.