A Blaschke-Petkantschin formula for linear and affine subspaces with application to intersection probabilities
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| Format: | Preprint |
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2024
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| author | Dare, Emil Kiderlen, Markus Thaele, Christoph |
| author_facet | Dare, Emil Kiderlen, Markus Thaele, Christoph |
| contents | Consider a uniformly distributed random linear subspace $L$ and a stochastically independent random affine subspace $E$ in $\mathbb{R}^n$, both of fixed dimension. For a natural class of distributions for $E$ we show that the intersection $L\cap E$ admits a density with respect to the invariant measure. This density depends only on the distance $d(o,E \cap L)$ of $L\cap E$ to the origin and is derived explicitly. It can be written as the product of a power of $d(o,E \cap L)$ and a part involving an incomplete beta integral. Choosing $E$ uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable $d(o,E \cap L)$ and study the behavior of the probability that $E \cap L$ hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where $E$ is tangent to the unit sphere, in which case we again derive the density for $d(o,E \cap L)$. Our probabilistic results are derived by means of a new integral-geometric transformation formula of Blaschke--Petkantschin type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_14253 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Blaschke-Petkantschin formula for linear and affine subspaces with application to intersection probabilities Dare, Emil Kiderlen, Markus Thaele, Christoph Metric Geometry Probability 52A22, 53C65, 60D05 Consider a uniformly distributed random linear subspace $L$ and a stochastically independent random affine subspace $E$ in $\mathbb{R}^n$, both of fixed dimension. For a natural class of distributions for $E$ we show that the intersection $L\cap E$ admits a density with respect to the invariant measure. This density depends only on the distance $d(o,E \cap L)$ of $L\cap E$ to the origin and is derived explicitly. It can be written as the product of a power of $d(o,E \cap L)$ and a part involving an incomplete beta integral. Choosing $E$ uniformly among all affine subspaces of fixed dimension hitting the unit ball, we derive an explicit density for the random variable $d(o,E \cap L)$ and study the behavior of the probability that $E \cap L$ hits the unit ball in high dimensions. Lastly, we show that our result can be extended to the setting where $E$ is tangent to the unit sphere, in which case we again derive the density for $d(o,E \cap L)$. Our probabilistic results are derived by means of a new integral-geometric transformation formula of Blaschke--Petkantschin type. |
| title | A Blaschke-Petkantschin formula for linear and affine subspaces with application to intersection probabilities |
| topic | Metric Geometry Probability 52A22, 53C65, 60D05 |
| url | https://arxiv.org/abs/2404.14253 |