Is there a reverse triangle inequality equivalent for $L^p$ spaces with $0I was trying to follow the proof for the reverse triangle inequality in the reals, but I got an inconsistency. Obviously, something is wrong, but I can't see what. Here is what I did:
In $L^p$ spaces with $0<p<1$, we have the reverse Minkowski's inequality:
$$||f+g||_p \geq ||f||_p + ||g||_p$$
So the following derivation is true:
$||f+g||_p - ||f||_p \geq ||g||_p$
Given that this is true for any $f,g \in L^p$, we can substitute $g$ with $g-f$ to get:
$||g||_p - ||f||_p \geq ||g-f||_p \tag{1}\label{1}$
On the other hand, from reverse Minkowski's inequality, we get:
$||f+g||_p - ||g||_p \geq ||f||_p$
Substituting $f$ with $f-g$ we get:
$||f||_p - ||g||_p \geq ||f-g||_p$
Multiplying by $-1$, we therefore obtain:
$||g||_p - ||f||_p \leq -||g-f||_p\tag{2}\label{2}$
Equations \ref{1} and \ref{2} are supposed to be true at the same time, but they are obviously inconsistent. However, I cannot find any mistake in my reasoning. What have I done wrong?

The reverse Minkowski inequality is not necessarily true for any $f$ and $g$. One needs extra assumptions. (For instance consider $f=-g$ with $\|f\|>0$. The reverse inequality says:
$$
0\geq 2\|f\|
$$ 
which is not true.)
A practical one is "$f$ and $g$ are non-negative (almost everywhere)". See Exercise 1 in this set of real analysis notes by Terry Tao. 
