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 Rⁿ, both of fixed dimension. For a natural class of distributions for E we show that the intersection L∩ E admits a density with respect to the invariant measure. This density depends only on the distance d(o,E ∩ L) of L∩ E to the origin and is derived explicitly. It can be written as the product of a power of d(o,E ∩ 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 ∩ L) and study the behavior of the probability that E ∩ 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 ∩ L). Our probabilistic results are derived by means of a new integral-geometric transformation formula of Blaschke--Petkantschin type.