# norms – What are the functions such that \$ lVert f + g rVert_p^p = lVert f rVert_p^p + lVert g rVert_p^p\$?

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