The $L_p$ Gauss image problem for <i>C</i> -pseudo-cones

Abstract In this article, we study the $L_p$ Gauss image problem for C -pseudo-cones, where C is a pointed closed convex cone in $\mathbb {R}^n$ with vertex at the origin and having nonempty interior. We first establish the existence of solutions for a Borel measure supported on compact subsets, and then prove a general existence theorem for $p&gt;0$ via approximation. In addition, we prove a uniqueness theorem for $p&lt;0$ under appropriate assumptions.