The $L_p$ Minkowski problem for $C$-close sets: existence and continuity

Let $C$ be a pointed closed convex cone in ${\mathbb{R}}^n$ with nonempty interior, and let $S^{n-1}$ denote the unit sphere in ${\mathbb{R}}^n$. The $L_p$ Minkowski problem for $C$-close sets is to determine, for a real number $p$ and a nonzero finite Borel measure $\mu$ defined on ${\mathrm{\Omega }}_{C^{\circ }}=S^{n-1}\cap \mathrm{i}\mathrm{n}\mathrm{t}{C}^{\circ }$, whether there exists a $C$-close set $\text{mathds}A$ such that $\mu$ is the $L_p$ surface area measure of $\text{mathds}A$. In this paper, we will solve the problem for $p\in (0,1)$ and for $\mu$ being a nonzero finite Borel measure on ${\mathrm{\Omega }}_{C^{\circ }}$. Moreover, we establish the continuity of solutions to the $L_p$ Minkowski problem for $p\in [0,1]$ in several settings.