In this talk, we establish the existence of horospherically convex hypersurfaces with prescribed shifted Gauss curvatures. This framework encompasses, as a special case, the Minkowski problem in hyperbolic space $H^{n+1}$. It is known that the support function of each horospherically convex hypersurface $M$ gives a conformal metric on the standard sphere $S^n$ whose Schouten curvatures differ only a constant with hyperbolic curvature radii of $M$. As a consequence, our results also yield solutions to a class of conformally invariant Nirenberg-type equations. This work is joint with Qi-Rui Li.