Grassmannian spines, projection closure operators, and diametric sweeps
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912868137435136 |
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| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| contents | 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 $p\in K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01514 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Grassmannian spines, projection closure operators, and diametric sweeps Chirvasitu, Alexandru Metric Geometry 51A05, 46C05, 51A20, 15A04, 51M15, 15A86 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 $p\in K$. |
| title | Grassmannian spines, projection closure operators, and diametric sweeps |
| topic | Metric Geometry 51A05, 46C05, 51A20, 15A04, 51M15, 15A86 |
| url | https://arxiv.org/abs/2602.01514 |