For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1, X_2\in K$.

Metric Geometry Probability Primary: 52A22, 52A40, 53C65, Secondary: 60D05

Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its $f$-vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14. 3 (1993), 381--394] in computational geometry.

Probability Primary 60D05, 60F05, Secondary 52A22, 53C65

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