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

TL;DR

This paper derives an explicit density formula for the intersection of random linear and affine subspaces in high-dimensional space, using a new integral-geometric transformation, with applications to intersection probabilities.

Contribution

It introduces a novel Blaschke-Petkantschin formula for linear and affine subspaces, enabling explicit density calculations for intersection points and probabilities.

Findings

Derived explicit density for intersection points with respect to the origin.

Analyzed the probability of intersection with the unit ball in high dimensions.

Extended results to tangent affine subspaces on the sphere.

Abstract

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.