On mappings, orthogonally additive in the Birkhoff-James sense (Q1076919)

Let \((X,\| \|)\) be a real normed linear space but not an inner product space of dimension \(\geq 2\) and \((G,+)\) be an arbitrary groupoid. Define the following binary relation \(\perp =\{(x,y)\in X\times X,\| x+\lambda y\| \geq \| x\|\) for all \(\lambda\in {\mathbb{R}}\}\). It is proved that a mapping F:X\(\to G\) is even and satisfies \[ (1)\quad F(x+y)=F(x)+F(y)\quad for\quad al\quad x,y\in X\quad with\quad x\perp y \] if and only if F is of the form \(F(x)=a_ 0\), if \(x=0\) and \(F(x)=a\), if \(x\neq 0\) where \(a_ 0,a\in G\) such that \(a_ 0+a_ 0=a_ 0\), \(a+a=a\) and \(a_ 0+a=a+a_ 0=a\). On the base of this result the author proves that if \((G,+)\) is an abelian group, then every function F:X\(\to G\) which satisfies (1) is additive.