Line-mappings preserving the area 1 of triangles (Q1916967)

The author studies the classical geometric problem of determining the conditions under which a line-preserving map \(\pi : E^n \to E^n\) of the Euclidean \(n\)-space to itself induces a collineation of the projective space \(\text{P}_2{\mathbb{R}}\). The map \(\pi\) is then called projective. For \(n \geq 3\), \textit{W. Benz} [`Geometrische Transformationen' (1992; Zbl 0754.51005)]proved Plücker's conjecture that \(\pi\) is projective if \(\pi\) is a bijection and two lines intersect iff their images under \(\pi\) do. \textit{J. Lester} [Aequationes Math. 28, 69-72 (1985; Zbl 0555.51010)]showed that \(\pi: E^3 \to E^3\) is projective if \(\pi\) is bijective and the distance of two lines is \(1\) iff the distance of their images under \(\pi\) is \(1\). Two years later [Util. Math. 31, 81-105 (1987; Zbl 0585.51015)]the same author proved that for \(n=2\) another suffucient condition is that \(\pi\) is bijective and preserves triangles of unit area (or perimeter or circumradius or inradius). In the first case it turns out that \(\pi\) is induced by an equiaffine mapping, while in the other cases \(\pi\) is associated with a motion. This result of J. Lester is the starting point of the present paper. The author is able to weaken the conditions considerably. Using elementary arguments he shows that \(\pi\) is projective if \(\pi\) merely preserves triangles of unit area. No restriction on \(n\) is made, but most part of the proof is spent for the case \(n=2\). While injectivity of \(\pi\) is easy to prove, it is hard to verify that \(\pi\) is surjective.