Kummer Configurations and S _m -Reflector Problems: Hypersurfaces in \({\mathbb{R}}^{n+1}\) with Given Mean Intensity

Abstract.

For a congruence of straight lines defined by a hypersurface in \({\mathbb{R}}^{n+1}\), n ≥ 1, and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions S _m , \(m = 1, 2, \ldots , n\), of principal intensities. The problem of existence and uniqueness of a closed hypersurface with prescribed S _n is the “reflector problem” extensively studied in recent years. In this paper we formulate and give sufficient conditions for solvability of an analogous problem in which the mean intensity S₁ is a given function.