An asymptotic rigidity property from the realizability of chirotope extensions

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 $\epsilon >0$, there is a finite, (generic) extension $\hat{P}$ 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 $\hat{P}$, then there exists a direct affine transform that maps each point of $Q$ within distance $\epsilon$ of the corresponding point of $P$.