  This paper continues the study of a class of compact convex hypersurfaces in
Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex
bodies obtained by taking the intersection of (solid) confocal paraboloids of
revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise
naturally in geometrical optics and are used in design of light reflectors and
reflector antennas. They are also important in rendering problems in computer
graphics.
  The notion of a focal function for reflectors plays a central role similar to
that of the Minkowski support function for convex bodies. In this paper the
basic question of when a given function is a focal function of a convex
reflector is answered by establishing necessary and sufficient conditions. In
addition, some smoothness properties of reflectors and of the associated
directrix hypersurfaces are also etablished.
