Bijections of \({\mathbb{R}}^ n\) onto itself (Q1108582)

Affine and continuous maps are characterized within the class of bijections of \({\mathbb{R}}^ n\) by the property that a certain set of geometric or topological figures is mapped into itself or another set of this type. A basic tool for the authors' characterization of affine maps is the fundamental theorem of projective geometry [\textit{G. Darboux}, Math. Ann. 17, 55-62 (1880)]. For example, affine maps are characterized by: each circle is mapped into the boundary of a strictly convex body or onto the boundary of a convex body (of any dimension). Also the existence of bijections of \({\mathbb{R}}^ n\) mapping circles into unions of two line segments, into quadrilaterals or onto unions of countably many line segments is shown. Continuity is implied by the following properties: i) circles are mapped onto simple closed curves \((n=2)\); ii) simple closed curves are mapped onto compact sets. The first result does not extend to \(n=3\). Finally in the infinite-dimensional case the preservation of circles is shown to characterize similarities.