We fix $1 \leq p < 2$. What are the couple $({x},{y})$ of vectors in $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$) for which the following equality holds \begin{equation} \label{eq:here} \lVert a {x} + b y \rVert_p^p = \lVert a {x} \rVert_p^p + \lVert b {y} \rVert_p^p \end{equation} for any $a,b \in \mathbb{R}$?
One easily sees that the condition is satisfied as soon as the Hadamard product ${x} \circ {y} = {0}$ since, in this case, $\lvert a x_i + b y_i \rvert^p= \lvert a x_i \rvert^p + \lvert b y_i \rvert^p$ for $i=1,2$ componentwise. My question is: is this sufficient condition also necessary? (I recall the assumption $p \neq 2$.)
I am also wondering the same question if we only know that the relation is satisfied for fixed $a,b$, for instance $a=b=1$. Are there more vectors satisfying the relation?