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 Poisson hyperplane process in the space, and consider the random polytope $Z_K$ defined as the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. If $f$ is a functional on convex bodies, then for increasing intensities of the process, the expectation of 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. We study these questions if $f$ is either the hitting functional or the mean width.

中文翻译：

---

凸体的泊松超平面过程和逼近

通过随机多胞体逼近 $\mathbb{R}^d$ 中的凸体 $K$ 的自然模型如下。取空​​间中的一个静止泊松超平面过程，并考虑随机多面体 $Z_K$ 定义为包含 $K$ 的所有封闭半空间的交集，这些半空间由不与 $K$ 相交的过程的超平面界定。如果 $f$ 是凸体上的泛函，那么随着过程强度的增加，差异 $f(Z_K)-f(K)$ 的期望可能会也可能不会收敛到零。如果是这样，那么收敛的顺序和可能的极限关系是有意义的。如果 $f$ 是击球函数或平均宽度，我们将研究这些问题。