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

TL;DR

This paper studies hypersurfaces in Euclidean space with prescribed mean intensity, introducing a new problem related to reflector design and providing conditions for the existence and uniqueness of solutions.

Contribution

It formulates a new reflector problem involving mean intensity and establishes sufficient conditions for its solvability.

Findings

Defined intensity in tangent directions for hypersurfaces

Introduced elementary symmetric functions of principal intensities

Provided solvability conditions for the mean intensity problem

Abstract

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ 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.