A Blaschke-Petkantschin formula for linear and affine subspaces with application to intersection probabilities

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.