On continuous expansions of configurations of points in Euclidean space

2011/07/01 by Holun Cheng, Ho-Lun Cheng, Ser Peow Tan +4
Computer Science · Mathematics · #51M16 #51M25 #52A20 #52A25 #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.MG #msc:51M16 #msc:51M25 #msc:52A20 #msc:52A25

paper · pdf · doi:10.48550/arxiv.1107.0140

8 pages, 4 figures

arxiv created 2011/07/01 · openalex publication_date 2011/07/01 · arxiv updated 2011/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any two configurations of ordered points p=(p₁,...,\p_N) and q=(q₁,...,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 ≥ 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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