On continuous expansions of configurations of points in Euclidean space

TL;DR

This paper demonstrates that the known continuous expansion in Euclidean space cannot be improved below dimension 2d for certain point configurations, establishing an optimality result with elementary techniques.

Contribution

It proves that for any dimension d ≥ 2, some point configurations require at least dimension 2d for a continuous expansion, showing the optimality of previous constructions.

Findings

Existence of configurations requiring dimension 2d for continuous expansion

Optimality of the known expansion dimension in Euclidean space

Elementary proof techniques used in the demonstration

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.