It is said in this Wikipedia article that the Lorentz space norm ${\|f\|_{L^{1,\infty }}}$ with ${f(x)=|x-1|^{-1}}$ is the area of the largest rectangle with sides parallel to the coordinate axes that can be inscribed in the graph.
However, direct calculation shows that the area of any inscribed rectangle is $$ \frac{(2-x)-x}{|1-x|}=2,\quad x\in(-\infty, 1). $$
The "largest rectangle" observation seems rather trivial. Is there something wrong with the Wikipedia interpretation?
The equality $\|f\|_{L^{1,\infty }}=\sup_{t>0} (t\{x:|f(x)|>t\})= t\{x:|f(x)|>t\}$ for any $t>0$ holds only for this special function $f$, not for an arbitrary function of $L^{1,\infty}$. So, I guess, the function $f$ can be a considered as an $L^{1,\infty}$ counterpart of a constant function $g(t)=c>0$ in $L^\infty$ for which $\|g\|_{ L^\infty}=\sup_t | g(t)|=g(t)$ for any $t$.