Kummer configurations and \(S_m\)-reflector problems: hypersurfaces in \({\mathbb{R}}^{n+1}\) with given mean intensity (Q2655742)

For a congruence of straight lines defined by a hypersurface in \(\mathbb R^{n+1}\), \(n \geq 1\), and a field of reflected directions created by a point source, the author defines the notion of intensity in a tangent direction and introduces some elementary symmetric functions \(S_m\), \(m=1,2,\dots,n\), of principal intensities. The well studied ``reflector problem'' is the problem of existence and uniqueness of a closed hypersurface with prescribed symmetric function \(S_n\). In this paper, the author formulates and gives sufficient conditions for the solvability of the problem in which the mean intensity \(S_1\) is a given function.