Grassmannian spines, projection closure operators, and diametric sweeps

For positive integers $r<d<n$ equip the powerset $2^{\mathbb{G}(r,V)}$ 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 $2r\le d$ the classification of closed subsets of $\mathbb{G}(r,V)$ rigidifies, these being precisely the sets of $r$-planes containing a fixed $(\le r)$-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 $p_0\in {\mathbb{R}}^{\mathbb{d}}$ as the compact fixed points of the dynamical system transforming $K\subseteq {\mathbb{R}}^d$ into its $p_0$-based diametric sweep: the union of all diameter-$p_{0}p$ balls for $p\in K$.