Let K be a convex set in R d and let K $λ$ be the convex hull of a homogeneous Poisson point process P $λ$ of intensity $λ$ on K. When K is a simple polytope, we establish scaling limits as $λ$ $\rightarrow$ $\infty$ for the boundary of K $λ$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $λ$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $λ$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity $\sqrt$ de dh dhdv.