On continuous expansions of configurations of points in Euclidean space

Abstract

For any two configurations of ordered points $p=(p_{1},...,\p_{N})$ and $q=(q_{1},...,q_{N})$ in Euclidean space $E^d$ such that $q$ is an expansion of $p$, there exists a continuous expansion from $p$ to $q$ in dimension 2d; Bezdek and Connelly used this to prove the Kneser-Poulsen conjecture for the planar case. In this paper, we show that this construction is optimal in the sense that for any $d \ge 2$ there exists configurations of $(d+1)^2$ points $p$ and $q$ in $E^d$ such that $q$ is an expansion of $p$ but there is no continuous expansion from $p$ to $q$ in dimension less than 2d. The techniques used in our proof are completely elementary.