Let $\ell_1^n$ be the space $\mathbb R^n$ equipped with the $L_1$, i.e., taxicab metric.
When $n=2$, a unit ball is a square with sides parallel to $45^\circ$.
When $n=3$, we get an octahedron, which looks like two pyramids glued together along their bottoms.
For general $n$, it's stated in various places that metric balls take the shape of an $n$-orthoplex, also called an $n$-cross-polytope, or also a hyperoctahedron.
However, I can't find a proof of this anywhere, though I did find many articles about various other aspects of so-called "taxicab geometry" like how the equivalent of $\pi$ is $4$, how to make ellipses, etc.
How can one show that taxicab balls take the form of an orthoplex?