Proposition: Directional inclusion

For nonempty compact convex sets $K,L$,

If $L$ is a polytope, its defining facet normals suffice. If true support functions have certified errors ${\delta }_K(n),{\delta }_L(n)$, the nominal gap $h_L(n)-h_K(n)\ge {\delta }_K(n)+{\delta }_L(n)$ certifies true inclusion.

Proof. Inclusion implies the inequalities. If $v\in K\setminus L$, choose its nearest point $z\in L$. Then $(v-z)\cdot (w-z)\le 0$ for all $w\in L$, which strictly separates $v$ from $L$. The facet and error statements follow by substitution.