On mappings preserving equilateral triangles

Abstract.

Let E be a Euclidean space, dim E ≥ 2. We say that f : E → E preserves equilateral triangles if for all triples of points x, y, z ∈ E $\|x-y\| = \|y-z\| = \|x-z\|$ we have $\|f(x)-f(y)\| = \|f(y)-f(z)\| =\|f(x)-f(z)\|.$ We show that if E is a finite-dimensional Euclidean space, dim E ≥ 2, f:E → E is measurable and preserves equilateral triangles, then it is a similarity transformation (an isometry multiplied by a positive constant). Moreover, in spaces at least three-dimensional we get a similarity transformation without any regularity assumption. Some generalizations as well as some interesting examples are also presented in the paper.