Summary
The following theorem holds true.
Theorem. Let X be a normed real vector space of dimension ⩽ 3 and let k > 0 be a fixed real number. Suppose that f: X → X and g: X × X → ℝ are functions satisfying ∥x − y∥ = k ⇒ f(x) − f(y) = g(x, y)⋅(x − y) for all x, y ∈ X. Then there exist elements λ ∈ ℝ and t ∈ X such that f(x) = λx + t for all x ∈ X and such that g(x, y) = λ for all x, y ∈ X with ∥x − y∥ = k.
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References
Benz, W. andBerens, H.,A contribution to a theorem of Ulam and Mazur. Aequationes Math.34 (1987), 61–63.
Benz, W.,The functional equation of dilatations on restricted domains. J. Univ. Kuwait Sci.14 (1987), 39–51.
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Benz, W. A conditional dilatation equation. Aeq. Math. 43, 177–182 (1992). https://doi.org/10.1007/BF01835699
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DOI: https://doi.org/10.1007/BF01835699