Grassmannian spines, projection closure operators, and diametric sweeps

For positive integers r<d<n equip the powerset 2^G(r,V) of the r-plane Grassmannian of an n-dimensional Hilbert space with the closure operator attaching to a set of r-planes the smallest superset which along with two r-planes also contains all r-dimensional orthogonal projections of one onto any d-plane containing the other. In the regime 2r≤d the classification of closed subsets of G(r,V) rigidifies, these being precisely the sets of r-planes containing a fixed (≤r)-plane. The result generalizes its (r,d,n)=(1,2,3) instance, of use in recent geometric-rigidity results motivated by matrix preserver problems. An auxiliary result classifies the balls centered at p₀∈R^d as the compact fixed points of the dynamical system transforming K⊆R^d into its p₀-based diametric sweep: the union of all diameter-p₀p balls for p∈K.