Beckman-Quarles type theorems for mappings from R^n to C^n
| dc.creator | Tyszka, Apoloniusz | |
| dc.date | 2003-01-14 | |
| dc.date.accessioned | 2026-07-07T04:54:27Z | |
| dc.date.available | 2026-07-07T04:54:27Z | |
| dc.description | Let G: C^n \times C^n -> C, G((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 G(x,y)=d^2 implies G(f(x),f(y))=d^2. Let A(n) denote the set of all positive numbers d such that any map f: R^n -> C^n that preserves unit distance preserves also distance d. Let D(n) denote the set of all positive numbers d with the property: if x,y \in R^n and |x-y|=d then there exists a finite set S(x,y) with {x,y} \subseteq S(x,y) \subseteq R^n such that any map f:S(x,y)->C^n that preserves unit distance preserves also the distance between x and y. We prove: (1) A(n) \subseteq {d>0: d^2 \in Q}, (2) for n>=2 D(n) is a dense subset of (0,\infty). Item (2) implies that each continuous mapping f from R^n to C^n (n>=2) preserving unit distance preserves all distances. | |
| dc.description | 13 pages, will appear in Aequationes Mathematicae | |
| dc.identifier | https://arxiv.org/abs/math/0301150 | |
| dc.identifier | http://arxiv.org/abs/math/0301150 | |
| dc.identifier | Aequationes Mathematicae 67 (2004), pp.225-235 | |
| dc.identifier | doi:10.1007/s00010-003-2719-1 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66257 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51M05 | |
| dc.title | Beckman-Quarles type theorems for mappings from R^n to C^n | |
| dc.type | text |