Rigidity, graphs and Hausdorff dimension
Abstract
For a compact set $E \subset \mathbb R^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$framework to be a collection of $k+1$ points in $E$ such that the distance between a pair of points is specified if the corresponding vertices of $G$ are connected by an edge. We regard two such frameworks as equivalent if the specified distances are the same. We show that in a suitable sense the set of equivalences of such frameworks naturally embeds in ${\mathbb R}^m$ where $m$ is the number of "essential" edges of $G$. We prove that there exists a threshold $s_k<d$ such that if the Hausdorff dimension of $E$ is greater than $s_k$, then the $m$dimensional Hausdorff measure of the set of equivalences of $G$frameworks is positive. The proof relies on combinatorial, topological and analytic considerations.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.05919
 Bibcode:
 2017arXiv170805919C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 42B10