In this talk, we will discuss the $L_p$ Brunn-Minkowski theory associated with the $C$-coconvex sets for $p\in [0,1)$ and the $L_p$ Minkowski problem associated with the $C$-coconvex sets for $p\in R$. In particular, we will talk about our solution to an open problem raised by Schneider in [Schneider, Adv. Math., 332 (2018), pp. 199-219]. This talk is based on a joint work with Jing Yang and Deping Ye.