Kummer configurations and $S_m-$reflector problems: Hypersurfaces in $\Rn$ with given mean intensity

Abstract

For a congruence of straight lines defined by a hypersurface in $R^{n+1},n\ge 1,$ 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 $S_m,m=1,2,...,n,$ of {\it 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 $S_1$ is a given function.

22 pages. This is a substantially revised and expanded version of an earlier posted paper