On mappings preserving equilateral triangles in normed spaces
Let \(\left(X, \| \cdot \| \right)\) and \(\left(Y, \| \cdot \| \right)\) be normed linear spaces with dimensions \(\geq 2\) and let \(f : X \longrightarrow Y\). We say that \(f\) preserves equilateral triangles if the condition \(\| x-y\| =\| y-z\| =\| z-x\| \) implies that \(\| f(x) - f(y)\| =\| f(y) - f(z) \| =\| f(z) - f(x)\| \), that \(f\) preserves isosceles triangles if the condition \(\| x-y\| =\| x-z\| \) implies that \(\| f(x) - f(y)\| =\| f(x) - f(z) \| \), and that \(f\) is a similarity if there exists \(k > 0\) such that \(\| f(x) - f(y)\| = k \| x - y \| \) for all \(x\) and \(y\). If one denotes these properties of \(f\) by (E), (I) and (S), respectively, then it is obvious that (S) \(\Longrightarrow\) (I) \(\Longrightarrow\) (E). The paper under review addresses the question whether the reverse implications hold, i.e., the question whether preserving a certain shape would imply preserving all shapes. This puts the paper in the category of the classical Beckman--Quarles theorem (with an elementary proof by \textit{W.\,Benz} in [Elem.\ Math.\ 42, 4--9 (1987; Zbl 0701.51013)]) and \textit{J.\,A.\,Lester}'s generalization in [J.~Geom.\ 27, 29--35 (1986; Zbl 0595.51021)], where the questions of whether preserving certain distances (respectively, areas, perimeters) would imply preserving all distances (respectively, areas, perimeters) are addressed. The paper under review proves that (i) if dim(\(X\)) \(\geq\) 3, then (E) \(\Longrightarrow\) (I), (ii) if dim(\(X\)), dim(\(Y\)) \(\geq\) 3, and if \(f\) is surjective, then (E) \(\Longrightarrow\) (S), and (iii) if \(X = Y\) is Euclidean with dim(\(X\)) = 2, and if \(f\) is non-constant and measurable, then (E) \(\Longrightarrow\) (S). The results above strengthen results proved earlier by the same authors in [J.~Geom.\ 80, 209--218 (2004; Zbl 1053.51012)], where they have treated the case when \(X = Y\) is Euclidean.
scientific article; zbMATH DE number 5118002