An asymptotic rigidity property from the realizability of chirotope extensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913848658755584 |
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| author | Goaoc, Xavier Padrol, Arnau |
| author_facet | Goaoc, Xavier Padrol, Arnau |
| contents | Let $P$ be a finite full-dimensional point configuration in $\mathbb{R}^d$. We show that if a point configuration $Q$ has the property that all finite chirotopes realizable by adding (generic) points to $P$ are also realizable by adding points to $Q$, then $P$ and $Q$ are equal up to a direct affine transform. We also show that for any point configuration $P$ and any $\varepsilon>0$, there is a finite, (generic) extension $\widehat P$ of $P$ with the following property: if another realization $Q$ of the chirotope of $P$ can be extended so as to realize the chirotope of $\widehat P$, then there exists a direct affine transform that maps each point of $Q$ within distance $\varepsilon$ of the corresponding point of $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An asymptotic rigidity property from the realizability of chirotope extensions Goaoc, Xavier Padrol, Arnau Combinatorics Computational Geometry Discrete Mathematics 52C40: Oriented matroids, 52C35: Arrangements of points, flats, hyperplanes, Let $P$ be a finite full-dimensional point configuration in $\mathbb{R}^d$. We show that if a point configuration $Q$ has the property that all finite chirotopes realizable by adding (generic) points to $P$ are also realizable by adding points to $Q$, then $P$ and $Q$ are equal up to a direct affine transform. We also show that for any point configuration $P$ and any $\varepsilon>0$, there is a finite, (generic) extension $\widehat P$ of $P$ with the following property: if another realization $Q$ of the chirotope of $P$ can be extended so as to realize the chirotope of $\widehat P$, then there exists a direct affine transform that maps each point of $Q$ within distance $\varepsilon$ of the corresponding point of $P$. |
| title | An asymptotic rigidity property from the realizability of chirotope extensions |
| topic | Combinatorics Computational Geometry Discrete Mathematics 52C40: Oriented matroids, 52C35: Arrangements of points, flats, hyperplanes, |
| url | https://arxiv.org/abs/2505.14189 |