Birkhoff-James orthogonality and Ratz's orthogonality

Question

Is Birkhoff-James orthogonality an orthogonality in the sense of Ratz?

Orthogonality in the sense of Ratz:

Suppose $X$ is a real vector space with $\dim X\geq2$ and $\perp$ is a binary relation on $X$ with the following properties:

1. Totality of $\perp$ for zero: $x\perp 0$ and $0\perp x$ for all $x$;

2. Independence: If $x,y\in X-\{0\}$ and $x\perp y$, then $x$ and $y$ are linearly independent.

3. Homogeneity: If $x,y\in X$ and $x\perp y$, then $\alpha x\perp\beta y$ for all $\alpha,\beta\in\mathbb{R}$.

4. Thalesian Property: If $P$ is a 2-dimensional subspace of $X$, $x\in P$ and $\lambda$ is a nonnegative real number, then $\exists$ $y_0\in P$ such that $x\perp y_0$ and $x+y_0\perp \lambda x-y_0$.

Birkhoff-James orthogonality:

$x$ is said to be Birkhoff-James orthogonal to $y$ if $\|x\|\leq\|x+\lambda y\|$ for all $\lambda\in\mathbb{R}$.

I could figure out that properties 1, 2 and 3 of Ratz's conditions for orthogonality are satisfied by Birkhoff-James orthogonality. The last condition bothers me.

Question. Is Birkhoff-James orthogonality an orthogonality in the sense of Ratz, or not?

Thanks in advance!

Answer 1

Yes, indeed, Birkhoff-James orthogonality is an orthogonality in the sense of Rätz. A proof appears on p.36 of

J. Rätz, On orthogonally additive mappings, Aequations Math. 28 (1985), 35-49.

The argument there is a refinement of the proof given on p.188 of

K. Sundaresan, Orthogonality and nonlinear functionals on Banach spaces. Proc. Amer. Math. Soc. 34 (1972), 187-190.

Since it is a non trivial proof, I just refer the interested reader to the above papers.