Mappings preserving \(B\)-orthogonality (Q1629135)

Recall that two vectors \(x\) and \(y\) in a real or complex normed space \(X\) are said to be Birkhoff orthogonal if \(\| x\|\leq \| x+\alpha y\|\) for every scalar \(\alpha\). In the papers [\textit{A. Koldobsky}, Proc. R. Soc. Edinb., Sect. A, Math. 123, No. 5, 835--837 (1993; Zbl 0806.46013)] and [\textit{A. Blanco} and \textit{A. Turnšek}, ibid. 136, No. 4, 709--716 (2006; Zbl 1115.46016)], it was proved that a nonzero linear map \(T\) from a normed space \(X\) into another normed space \(Y\) preserves Birkhoff orthogonality if, and only if, there is some \(c>0\) such that \(\| Tx\|=c\| x\|\) for all \(x\in X\). The author gives a simpler proof of the above result. Also, he proves that the above result is also true, in the real case, assuming only that the map \(T\) is additive, the linearity of \(T\) being a consequence.