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A natural model for the approximation of a convex body $K$ in $\mathbb{R}^d$by random polytopes is obtained as follows. Take a stationary Poissonhyperplane process in the space, and consider the random polytope $Z_K$ definedas the intersection of all closed halfspaces containing $K$ that are bounded byhyperplanes of the process not intersecting $K$. If $f$ is a functional onconvex bodies, then for increasing intensities of the process, the expectationof the difference $f(Z_K)-f(K)$ may or may not converge to zero. If it does,then the order of convergence and possible limit relations are of interest. Westudy these questions if $f$ is either the hitting functional or the meanwidth.

Poisson hyperplane processes and approximation of convex bodies