does this convex set have a specific name?

Question

Let $x_1,\dots,x_N$ be points of $\mathbb{R}^n$. Define the following set: $\mathcal{A} = \left\{\sum_{j=1}^N a_j x_j : -1 \le a_j \le 1, \, \, \forall j=1,...,N\right\}$. It is an easy exercise to see that $\mathcal{A}$ is a convex set. Does this set have any specific name along the lines of convex/affine/linear hull?

Answer 1

Zonotope.

(I have nothing more to say, but say more to satisfy the computer.)

Answer 2

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\setA}{\mathcal{A}}$If the points $(x_{j})_{j=1}^{N}$ are linearly independent as vectors in $\Reals^{n}$, the set $\setA$ is the parallelipiped[1] centered at the origin, with a vertex at $-\sum_{j} x_{j}$ and edges $(2x_{j})_{j=1}^{N}$.[2]

If the points are linearly dependent, $\setA$ is a projection of a parallelipiped.[3]

In any case, $\setA$ is the image of the cube $[-1, 1]^{N}$ under the linear transformation $T:\Reals^{N} \to \Reals^{n}$ whose standard matrix has $x_{j}$ as $j$th column.

1. According to Wikipedia, Coxeter called such a set a parallelotope. I seem to recall hearing the term "parallelipiped" used generically, as in "a fundamental domain for an Abelian variety is a parallelipiped in $\mathbf{C}^{n}$".

2. If $p_{0} \in \Reals^{n}$ and $(x_{j})_{j=1}^{N}$ is a linearly independent set of vectors, the associated parallelipiped is defined to be the set of linear combinations $p_{0} + \sum_{j} t_{j} x_{j}$ for which $0 \leq t_{j} \leq 1$ for all $j$.

3. I've never encountered a special name for such sets, but I don't work in convex geometry.

(Wikifying this answer since (i) in part this answer incorporates comments made by user208259 and p.s., and (ii) terminology may differ between various branches of geometry, and convex geometers passing by in the future should feel free to make edits.)