A conditional dilatation equation

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.