aequationes mathematicae

, Volume 43, Issue 2–3, pp 177–182 | Cite as

A conditional dilatation equation

  • Walter Benz
Research Papers


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.

AMS (1980) subject classification

Primary 39B40, 39B70 Secondary 51A10 


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  1. [1]
    Benz, W. andBerens, H.,A contribution to a theorem of Ulam and Mazur. Aequationes Math.34 (1987), 61–63.Google Scholar
  2. [2]
    Benz, W.,The functional equation of dilatations on restricted domains. J. Univ. Kuwait Sci.14 (1987), 39–51.Google Scholar

Copyright information

© Birkhäuser Verlag 1992

Authors and Affiliations

  • Walter Benz
    • 1
  1. 1.Mathematisches SeminarUniversität HamburgHamburg 13Germany

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