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A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces

Apoloniusz Tyszka
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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.

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Apoloniusz Tyszka. 2000-10-08. A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces. https://arxiv.org/abs/math/0008135

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