Poisson polytopes

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope ${\mathrm{\Pi }}_{\eta }$ in a fixed convex polytope $P\subset {\mathbb{R}}^d$. Here, ${\mathrm{\Pi }}_{\eta }$ is the convex hull of the intersection of a Poisson process $X$ of intensity $\eta$ with $P$.