The Beckman-Quarles theorem for continuous mappings from R^n to C^n
| dc.creator | Tyszka, Apoloniusz | |
| dc.date | 2002-04-13 | |
| dc.date | 2002-04-24 | |
| dc.date.accessioned | 2026-07-07T04:47:40Z | |
| dc.date.available | 2026-07-07T04:47:40Z | |
| dc.description | Let ϕ((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that f:R^n -> C^n preserves distance d>=0 if for each x,y \in R^n ϕ(x,y)=d^2 implies ϕ(f(x),f(y))=d^2. We prove that if x,y \in R^n (n>=3) and |x-y|=(\sqrt{2+2/n})^k \cdot (2/n)^l (k,l are non-negative integers) then there exists a finite set {x,y} \subseteq S(x,y) \subseteq R^n such that each unit-distance preserving mapping from S(x,y) to C^n preserves the distance between x and y. It implies that each continuous map from R^n to C^n (n>=3) preserving unit distance preserves all distances. | |
| dc.description | 9 pages, added proofs of technical lemmas | |
| dc.identifier | https://arxiv.org/abs/math/0204171 | |
| dc.identifier | http://arxiv.org/abs/math/0204171 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63807 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51M05 (Primary) | |
| dc.title | The Beckman-Quarles theorem for continuous mappings from R^n to C^n | |
| dc.type | text |