An asymptotic rigidity property from the realizability of chirotope extensions

TL;DR

This paper establishes a rigidity property for point configurations in Euclidean space based on the realizability of their chirotope extensions, showing that certain extension properties imply affine equivalence or approximate congruence.

Contribution

It proves that configurations with identical realizability of all their chirotope extensions are affinely equivalent, and introduces an asymptotic rigidity result for approximate realizations.

Findings

Configurations with identical chirotope extension realizability are affine equivalents.

Existence of approximate extensions implies near-affine congruence.

Provides a new perspective on the rigidity of point configurations via chirotopes.

Abstract

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 $P$.