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?

  • $\begingroup$ I'm not sure if there's any particular name, but this describes a Minkowski sum over some line segments (which have the origin as their midpoint, but any sum of line segments would be equal, up to translation, to this) $\endgroup$ Jan 21 '15 at 2:30
  • 3
    $\begingroup$ It's the image under a linear mapping of the hypercube $[-1,1]^N$. Consequently it's the convex hull of the $2^N$ points of the form $\sum \varepsilon_i x_i$, where $\varepsilon_i \in \{-1,1\}$. $\endgroup$
    – user208259
    Jan 21 '15 at 2:35
  • $\begingroup$ @user208259: That sounds interesting... $\endgroup$
    – Manos
    Jan 21 '15 at 2:37
  • $\begingroup$ The convex hull of points $z_1, \dots, z_p$ is the set of points of the form $\sum c_i z_i$ where $\sum c_i = 1$ and $c_i \geq 0$. The image of this set under a linear mapping $f$ is precisely the convex hull of the points $f(z_i)$. $\endgroup$
    – user208259
    Jan 21 '15 at 3:03
  • 1
    $\begingroup$ When $n \ge N$ it's a N-dimensional parallelotope. When $n < N$ it's the projection of a parallelotope. $\endgroup$
    – p.s.
    Jan 23 '15 at 1:34


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

  • $\begingroup$ Minkowski sum of line segments, projection of a cube... Yeah, that's a zonotope! $\endgroup$
    – pjs36
    Apr 25 '15 at 14:38

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\setA}{\mathcal{A}}$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.)


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