Mappings preserving some geometrical figures

Abstract

We introduce new characterizations of linear isometries. More precisely, we prove that if a one-to-one mapping f : Rn →Rn (n > 1) 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 : Rn →Rn up to translation such that f(x)=(b/a) I(x).