A Characterization of Similarity Maps Between Euclidean Spaces Related to the Beckman--Quarles Theorem

Details A Characterization of Similarity Maps Between Euclidean Spaces Related to the Beckman--Quarles Theorem A Characterization of Similarity Maps Between Euclidean Spaces Related to the Beckman--Quarles Theorem

2002-03-22

arxiv.org/abs/math/0203233v1

Mathematics

Metric Geometry

5 pages

Scientific paper

It is shown that each continuous transformation $h$ from Euclidean $m$-space
($m>1$) into Euclidean $n$-space that preserves the equality of distances (that
is, fulfils the implication $|x-y|=|z-w|\Rightarrow|h(x)-h(y)|=|h(z)-h(w)|$) is
a similarity map. The case of equal dimensions already follows from the
Beckman--Quarles Theorem.

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