Abstract We study Funk-type transforms associated with intersections of the unit sphere in R n with lower-dimensional affine planes passing through a given point inside or outside the sphere. Our goal is to investigate injectivity of such “paired” transforms generated by two families of planes centered at distinct points. Necessary and sufficient conditions of injectivity are obtained in terms of geometry of the centers location. The technique used is based upon the action of the automorphism group of the unit ball and a related billiard-like dynamics on the sphere.

中文翻译：

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非中心 Funk-Radon 变换：单个和多个

摘要 我们研究了与 R n 中的单位球面与穿过球内或球外给定点的低维仿射平面相交相关的 Funk 型变换。我们的目标是研究由以不同点为中心的两个平面家族产生的这种“成对”变换的注入性。就中心位置的几何形状而言，获得了注入性的必要和充分条件。所使用的技术基于单位球的自同构群的作用以及球体上相关的类似台球的动力学。