A duality for Poisson flats

In keeping with the intersection density of a stationary Poisson process of r-flats in Euclidean d-space, where r >= d/2, we introduce a notion of closeness, called proximity, for such processes if r < d/2. It is shown that the two notions are connected by a duality: the proximity of a stationary Poisson r-flat process is, up to a constant factor, the intersection density of a certain unique stationary Poisson (d-r)-flat process.