Zonotope

Remarks

Equivalently, $Z = \set{c + \lambda_1 v_1 + \cdots + \lambda_m v_m : 0 \le \lambda_i \le 1}$ with $c = a_1 + \cdots + a_m$ and $v_i = b_i - a_i$. So $Z$ is the image of the cube $[0,1]^m$ under the affine map $\lambda \mapsto c + \sum_i \lambda_i v_i$, and $Z$ is a convex polytope. Every zonotope is centrally symmetric about the point $c + \tfrac12 (v_1 + \cdots + v_m)$.

Example: in $\R^2$, take $v_1 = (1, 0)$, $v_2 = (\tfrac12, \tfrac{\sqrt3}{2})$, $v_3 = (-\tfrac12, \tfrac{\sqrt3}{2})$ and $c = 0$. The sum of the three segments $[0, v_i]$ is a regular hexagon with side length $1$.

Non-example: a triangle is not a zonotope, because it is not centrally symmetric.