二维严格凸赋范空间中Beckman-Quarles定理的离散形式
A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces
摘要：
设X是实赋范向量空间，且dim X \ge 2.设d>0是一个固定的实数。我们证明，如果x，y \in X和||X-Y||/d是有理数则存在有限集{x，y} \subseteq S（x，y）\subseteq X具有以下性质：对于每个2维的严格凸Y，每个从S（x，y）到Y的保持距离d的映射保持x和y之间的距离。这意味着从X到Y的每个保持距离d的映射都是等距的。
英文摘要：
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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