scholarly journals Reporting Neighbors in High-Dimensional Euclidean Space

2014 ◽  
Vol 43 (4) ◽  
pp. 1363-1395 ◽  
Author(s):  
Dror Aiger ◽  
Haim Kaplan ◽  
Micha Sharir
Keyword(s):  
Euclidean Space ◽  
High Dimensional ◽  
Dimensional Euclidean Space

scholarly journals Reflection-Like Maps in High-Dimensional Euclidean Space

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 872
Author(s):  
Zhiyong Huang ◽  
Baokui Li
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Orthogonal Transformation ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Domain ◽  
Euclidean Spaces ◽  
Line Cross ◽  
Klein Model

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

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