On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet}=C\setminus A$ is a closed convex set. For $0<p<1$, the $p$-co-sum of $C$-coconvex sets is introduced, and the corresponding $L_p$ Brunn-Minkowski inequality for $C$-coconvex sets is established. We also define the $L_p$ surface area measures, for $0\neq p\in \mathbb{R}$, of certain $C$-coconvex sets, which are critical in deriving a variational formula of the volume of the Wulff shape associated with a family of functions obtained from the $p$-co-sum. This motivates the $L_p$ Minkowski problem aiming to characterize the $L_p$ surface area measures of $C$-coconvex sets. The existence of solutions to the $L_p$ Minkowski problem for all $0\neq p\in \mathbb{R}$ is established. The $L_p$ Minkowski inequality for $0<p<1$ is proved and is used to obtain the uniqueness of the solutions to the $L_p$ Minkowski problem for $0<p<1$. For $p=0$, we introduce $(1-\tau)\diamond A_1\oplus_0\tau\diamond A_2$, the log-co-sum of two $C$-coconvex sets $A_{1}$ and $A_{2}$ with respect to $\tau \in(0, 1)$, and prove the log-Brunn-Minkowski inequality of $C$-coconvex sets. The log-Minkowski inequality is also obtained and is applied to prove the uniqueness of the solutions to the log-Minkowski problem that characterizes the cone-volume measures of $C$-coconvex sets. Our result solves an open problem raised by Schneider in [Schneider, Adv. Math., 332 (2018), pp. 199-219].