Grassmannian spines, projection closure operators, and diametric sweeps
Alexandru Chirvasitu
TL;DR
The paper introduces a saturation framework on Grassmannians via a ternary relation $\\tau_d$ that captures how $r$-planes project into $d$-planes. It proves a sharp rigidity result: when $1<d<n$, the only $\\tau_d$-saturated subsets of the projective space are singletons or the whole space, while the $r$-plane spines (sets of all $r$-planes containing a fixed core) exhaust the saturated families, establishing a bijection between ${\\mathbb G}(\\le r,V)$ and saturated $r$-plane sets. An auxiliary diameter-sweep dynamic identifies fixed points as $p_0$-centered balls, linking geometric closure behavior to a simple, canonical fixed point structure. These results generalize known low-dimensional cases connected to matrix preserver problems and relate to broader rigidity phenomena in projective geometry and Wigner-type theorems.
Abstract
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 $p_0\in \mathbb{R^d}$ 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-$p_0p$ balls for $p\in K$.