  We show that if the Gauss Image Measure of submeasure $\lambda$ via convex
body $K$ agrees with the Gauss Image Measure of $\lambda$ via convex body $L$,
then the radial Gauss Image maps of their duals, are equal to each other almost
everywhere as multivalued maps with respect to $\lambda$. As an application of
this result, we establish that, in this case, dual bodies, $K^*$ and $L^*$, are
equal up to a dilation on each rectifiable path connected component of the
support of $\lambda$. Additionally, we provide many previously unknown
properties of the radial Gauss Image map, most notably its variational
Lipschitz behavior, establish some measure theory concepts for multivalued maps
and, as a supplement, show how the main uniqueness statement neatly follows
from the Hopf Theorem under additional smooth assumptions on $K$ and $L$.
