Polyhedral uncertainty set ($\mathcal{U}=\{r:\Vert P(r-r_0){\Vert }_{\infty }\le 1\}$)

A polyhedral uncertainty set is specified by finitely many linear inequalities. The displayed centered example is the inverse image of a box under a linear map. It is nonempty because it contains $r_0$, and contains every line $r_0+td$ with $d\in \ker P$. Consequently it need not be bounded when $P$ is rank deficient. Its supremum norm constraint is equivalent to $-1\le P(r-r_0)\le 1$.