Let $1 leq p leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g in L_p(mathbb{R})$ such that

$$ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p.$$

For $p = 2$, this relation is satisfied if and only if $langle f, g rangle = 0$. For $p = 1$, it has been shown in this post that the condition is equivalent to $f g geq 0$ almost everywhere.

For a general $p$, the relation is clearly satisfied as soon as the product $fg=0$ almost everywhere. Is this latter sufficient condition also necessary?

If not, then what if we reinforce the condition with

$$ lVert alpha f + beta g rVert_p^p = lVert alpha f rVert_p^p + lVert beta g rVert_p^p$$

for any $alpha, beta in mathbb{R}$?