Name of the geometric figure of points ${\bf x} \in \Bbb R^n$ with $1$-norm $||{\bf x}||_1 = 1$

Question

Is there a name for the figure

$$\{{\bf x} \in \Bbb R^n : ||{\bf x}||_1 = 1\} \subset \Bbb R^n ?$$

Things like this seem to usually have names, for instance, the $n$-cube or $n$-ball.

In $2$ dimensions it would simply be (the boundary of) a rotated square, and in $3$ dimensions a suitably positioned octahedron with 'radius' $1$.

Answer 1

The $n$-dimensional generalization of an octahedron is called cross-polytope.