Mappings preserving some geometrical figures (Q1878580)

The author presents the following new characterization of linear isometries of \(\mathbb R^n\) \((n>1)\): If a one-to-one mapping \(f:\mathbb R^n\to \mathbb R^n\) maps the periphery of every regular triangle (quadrilateral or hexagon) of side length \(a>0\) onto the periphery of a figure of the same type with side length \(b>0\), then there exists a linear isometry \(I:\mathbb R^n\to \mathbb R^n\) up to translation such that \(f=(b/a)I\).