 Support function of an inverse image of an infinity-norm ball

If $x\notin \mathrm{range}{P}^T=(\ker P{)}^{\perp }$, the support function is infinite along a kernel line. Otherwise linear programming duality gives the displayed minimum, attained because the corresponding linear programs are feasible with finite values. Splitting $q=\alpha -\beta$ into nonnegative parts yields the equivalent minimum of $1^T(\alpha +\beta )$. Thus worst-case revenue over a polyhedral uncertainty set is $r_0^{T}x-{\sigma }_C(x)$, with value $-\infty$ off the row space.