The integral geometry boundary determination problem for a pencil of straight lines

Under consideration is the problem of integrating finitely many functions over straight lines. Each function as well as the corresponding line is assumed unknown. The available information is the sum of integrals over all straight lines of a family of pencils in each of which the intersection of lines is a point of a given bounded open set in a finite-dimensional Euclidean space. Each integrand depends on a greater number of variables than the sum of the integrals. Hence, the conventional statement of the problem of determining the integrands becomes underspecified. In this situation we pose and study the problem of determining the discontinuity surfaces of the integrands. The uniqueness theorem is proven under the condition that these surfaces exist. The present article is a refinement of the previous studies of the authors and differs from them in [1–6] by not only some technical improvements but also the principally new fact that the integration is performed over an unknown set.