Injectivity of pairs of non-central Funk transforms

TL;DR

This paper investigates the injectivity of pairs of Funk transforms on the sphere, focusing on how the geometric configuration of centers affects the transforms' ability to uniquely recover functions.

Contribution

It provides necessary and sufficient conditions for the injectivity of paired Funk transforms based on the positions of their centers, using automorphism group actions and billiard dynamics.

Findings

Injectivity depends on the geometric configuration of centers.

Necessary and sufficient conditions for injectivity are established.

Automorphism group actions and billiard dynamics are key tools.

Abstract

We study Funk-type transforms on the unit sphere in R^n associated with cross-sections of the sphere by lower-dimensional planes passing through an arbitrary fixed point inside the sphere or outside. Our main concern is injectivity of the corresponding paired transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions for the paired transforms to be injective are obtained, depending on geometrical configuration of the centers. Our method relies on the action of the automorphism group of the unit ball and the relevant billiard-like dynamics on the sphere.