The integral geometry problem for a family of cones in the \(n\)-dimensional space (Q1358033)

Let \(\{K(x,y)\}\) be the family of the cones having as vertices the points \((x,y)\) and which are determined by the relations \[ \sum^{n-1}_{m=1}(x_m-\xi_m)^2=(y-\eta)^2,\quad 0\leq \eta \leq y, \] where \(x=(x_1,x_2,\dots,x_{n-1})\in\mathbb{R}^{n-1}\), \(\xi=(\xi_1,\xi_2,\dots,\xi_{n-1})\in\mathbb{R}^{n-1}\), \(n>2\), \(y\in \mathbb{R}^1\), \(\eta\in \mathbb{R}^1\), \(y\geq 0\), \(\eta\geq 0\) and \(Q(x,y)\) is the part of the \(n\)-dimensional space which is bounded by the surface of the cone \(K(x,y)\) and the hyperplane \(y=0\). Denote by \(d\kappa\) and \(dq\) the area element on \(K\) and the volume element of \(Q\), respectively. The author proves uniqueness theorems for the integral equations \[ \iint_{K(x,y)} \kappa(\xi,\eta)d\kappa=f(x,y)\tag{1} \] and \[ \iint_{K(x,y)} u(\xi,\eta)d\kappa+ \iiint_{Q(x,y)} g(x,y,\xi,\eta) u(\xi,\eta)dq=F(x,y)\tag{2} \] in the case of an even-dimensional space and constructs a simple representation for a solution to (1). He obtains stability estimates for solutions to (1) and (2) in the Sobolev spaces and shows their weak incorrectness.