Grassmannian spines, projection closure operators, and diametric sweeps

TL;DR

This paper studies the structure of certain closure operators on Grassmannian sets, classifies closed subsets in specific regimes, and introduces a dynamical system related to diametric sweeps with applications in geometric rigidity.

Contribution

It provides a classification of closed subsets of Grassmannians under a new closure operator and introduces a diametric sweep dynamical system with fixed point characterization.

Findings

Closed subsets are characterized as sets containing a fixed subspace in the regime 2r ≤ d.

The classification generalizes known cases, including (r,d,n)=(1,2,3).

Balls centered at a point are fixed points of the diametric sweep transformation.

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 $p\in K$.