= Solution
For fixed $x$, put $z=r-r_0$. The <polyhedral uncertainty set> is equivalent to $-\mathbf1\le Pz\le\mathbf1$, so the inner problem is the <linear program>
$$
\boxed{\inf_{z\in\mathbb R^n}\ r_0^Tx+x^Tz\quad\text{subject to }Pz\le\mathbf1,\ -Pz\le\mathbf1.}
$$
The variable $z$ is unrestricted in sign. The point $z=0$ is strictly feasible. If the optimum is finite, it is attained, so the infimum is a minimum. The use of an infimum also covers possible unbounded cases permitted by the printed assumptions; strict feasibility alone does not guarantee a finite objective.
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