A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces
Functional Analysis
2007-05-23 v5 Metric Geometry
Abstract
Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real number. We prove that if x,y \in X and ||x-y||/d is a rational number then there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following property: for each strictly convex Y of dimension 2 each map from S(x,y) to Y preserving the distance d preserves the distance between x and y. It implies that each map from X to Y that preserves the distance d is an isometry.
Keywords
Cite
@article{arxiv.math/0008135,
title = {A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces},
author = {Apoloniusz Tyszka},
journal= {arXiv preprint arXiv:math/0008135},
year = {2007}
}
Comments
LaTeX 2.09, with a note that S(x,y) does not depend on Y