Show that a set is a polyhedron

Question

How can I show that the set

$$S=\{y_1a_1 +y_2a_2 ~| −1\leq (y_1 \land y_2) \leq 1\}$$

where $a_1, a_2 \in \mathbb R^n$ are given, is a polyhedron?

Answer 1

The set $S$ is the convex hull of the four points $\pm a_1 \pm a_2$, hence a polyhedron:

Set $\lambda = \frac{y_1+1}2$, $\mu=\frac{y_2+1}2$. Then $$ y_1a_1+y_2a_2 = \lambda\mu(a_1+a_2) + \lambda(1-\mu) (a_1-a_2) + (1-\lambda)\mu(-a_1+a_2)+(1-\lambda)(1-\mu)(-a_1-a_2). $$