Point and line mappings preserving volume 1 of \(n\)-dimensional simplices in \(\mathbb{R}^ n\) (Q1904254)

The author proves the following two results about mappings in \(\mathbb{R}^n\), \(n \geq 3\). (i) A mapping \(f\) of \(\mathbb{R}^n\) into itself, such that \(f(x_1),\dots, f(x_{n+1})\) are the vertices of a simplex of volume 1 if \(x_1,\dots,x_{n+1}\) are the vertices of a simplex of volume 1, must be equiaffine. (As the author remarks, this is also true if \(n = 2.\)) (ii) Let \(M^n\) be the set of lines of \(\mathbb{R}^n\). A map \(\pi\) of \(M^n\) into itself, such that \(\pi(a_1), \dots, \pi(a_{{1\over 2}n(n+1)})\) are the edges of a simplex of volume 1 if \(a_1,\dots,a_{{1\over 2}n(n+1)}\) are the edges of a simplex of volume 1, must be induced by an equiaffine mapping of \(\mathbb{R}^n\).