Abstract:
We consider the problem of the distribution of the number of intersection points of $n$ random hyperplanes inside a convex body $D\subset\mathbb{R}^d$. A new approach is proposed for studying the probabilities $p_{nk}^{(d)}$ that exactly $k$ out of ${n\choose d}$ possible intersection points of $d$-dimensional subsets of hyperplanes lie inside $D$. The method is based on expressing the desired probabilities in terms of expected values of products of indicator functions and applying Stirling numbers of the second kind. A general formula for $p_{nk}^{(d)}$ is obtained in terms of these expectations. For the case $n=d+1$, explicit expressions for the probabilities are found in terms of new geometric invariants of the convex body, generalizing classical results for the planar case $(d=2)$ to arbitrary dimension.
Key words and phrases:random hyperplanes, convex body, Crofton's measure, geometric probability, Stirling numbers of the second kind, intersection points, integral-geometric invariants, integral geometry.
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