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arXiv · 1107.0140

On continuous expansions of configurations of points in Euclidean space

Holun Cheng·Ser Peow Tan·Yidan Zheng
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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.

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Holun Cheng, Ser Peow Tan, Yidan Zheng. 2011-07-01. On continuous expansions of configurations of points in Euclidean space. https://arxiv.org/abs/1107.0140

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