The Uniqueness of the Gauss Image Measure

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, $K^{\ast }$ and $L^{\ast }$, 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 $L$.