Mappings preserving area 1 of triangles (Q1879088)

Let \((X, \langle , \rangle)\) be a real inner product space, and let \(\| \| \) be the euclidean norm on \(X\) defined by \(\| x \| = \langle x,x \rangle\). The euclidean area \(\Delta (a,b,c)\) of a triangle \(a\), \(b\), \(c\) is defined by \([\langle a-c, b-c \rangle]^2 = \| a-c\| ^2 \| b-c\| ^2 - 4[\Delta(a,b,c)]^2.\) A function \(f : X \longrightarrow X\) is called distance 1 preserving if \(\| f(a) -f(b)\| = 1\) for all \(a, b \in X\) with \(\| a -b\| = 1\), and is called area 1 preserving if \(\Delta (f(a), f(b), f(c)) = 1\) for all \(a, b, c \in X\) with \(\Delta (a,b,c) = 1\). The classical Beckman-Quarles theorem (with an elementary proof by \textit{W. Benz} [Elem. Math. 42, 4--9 (1987; Zbl 0701.51013)]) states that if \(2 \leq \dim X < \infty\), then every distance 1 preserving function \(f\) on \(X\) is of the form \[ f(x) = \omega (x) + t, \] where \(\omega : X \longrightarrow X\) is an orthogonal linear mapping, and where \(t \in X\). A beautiful analogue by \textit{J. Lester} in [J. Geom. 27, 29--35 (1986; Zbl 0595.51021)] states that if \(3 \leq \dim X < \infty\), then every area 1 preserving function on \(X\) is of that form. In this paper, the author provides simple and elegant examples showing that the assumption, in Lester's theorem, that dim(\(X\)) \(< \infty\) is irredundant, and that it can be replaced by the assumption that there exists an equilateral triangle \(a, b, c \in X\) of side length \(\sqrt{2}\) with \(\Delta (f(a), f(b), f(c)) = \Delta (a,b,c)\).