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 : R(n) --> R(n) (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 : R(n) --> R(n) up to translation such that f (x) = (b/a)I(x).