2024/04/22 by Dare, Emil, Kiderlen, Markus, Thaele, Christoph
#52A22 #53C65 #60D05 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2404.14253
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.