On a measure space $(X,\mu)$, for $0<p,q<\infty$ the Lorentz space $L^{p,q}(\mu)$ is defined by $$\|f\|_{L^{p,q}(\mu)}:=p^\frac1q\|t\cdot\mu(|f|>t)^\frac1p\|_{L^q(\mathbb R_+,\frac{dt}t)}=p^\frac1q\left(\int_0^\infty t^{q-1}\mu(|f|>t)^\frac qpdt\right)^{1/q},$$ with usual modification for $q=\infty$: $$\|f\|_{L^{p,\infty}(\mu)}:=\|t\cdot\mu(|f|>t)^\frac1p\|_{L^\infty(\mathbb R_+,\frac{dt}t)}=\sup_{t>0}t\cdot\mu(|f|>t)^\frac 1p.$$
Although for $1<p,q<\infty$, the space $L^{p,q}(\mu)$ is normable, but the quantatity $\|\cdot\|_{L^{p,q}(\mu)}$ itself may only be a quasi-norm.
I want to determine all $(p,q)$ such that $\|\cdot\|_{L^{p,q}(\mu)}$ is a norm, i.e. $\|f+g\|_{L^{p,q}(\mu)}\le \|f\|_{L^{p,q}(\mu)}+\|g\|_{L^{p,q}(\mu)}$ for all $f,g$.
In Grafakos' Classical Fourier analysis, he mentioned the example $f(x)=x$, $g(x)=1-x$ on the space $x\in[0,1]$. This only gives a partial result where $\|\cdot\|_{L^{p,q}}$ is not a norm. For more general case I haven't found a reference about this.
Lorentz himself proved that $\|\cdot\|_{L^{p,q}}$ is a norm if and only if $1\leq q\leq p<\infty$ (G.G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), 411–429). I don't know much about those spaces myself, but I found this result, along with other nice ones, neatly summarized in the introduction of this paper (S. Barza, V. Kolyada, J. Soria, Sharp constants related to the triangle inequality in Lorentz spaces (2007)). Basically it seems the decreasing rearrangement form of this (quasi-)norm tends to be preferred, as it's much simpler to handle and doesn't even end up depending on $\mu$.