In a normed space $X$, the Birkhoff orthogonality is defined as following: $$x\perp_B y \quad \Leftrightarrow \quad \| x+\lambda y\| \geq \| x\|.$$
A well known characterization of the Birkhoff orthogonality is: $$x\perp_B y \quad \Leftrightarrow \quad \exists \varphi \in \{ \varphi\in X^* \; :\; \| \varphi\| =1, \; \varphi (x)=\| x\|\} \; : \; \varphi (y)=0.$$
A normed linear space X is called smooth if to each $x\in X$, $x\neq 0$, there corresponds a unique Hahn-Banach functional $\varphi \in X^*$ satisfying $\| \varphi\| =1$ and $\varphi (x)=\| x\|$.
Let $X$ be a real smooth normed space. I was wondering if someone could tell me why smoothness of $X$ yields the Birkhoff orgonality is additive on the right: $x\perp_B y$ and $x\perp_B z$ implies $x\perp_B y+z$?
Thannks in advance.