Proposition: Directional certificate

For compact convex $K,L\subset {\mathbb{R}}^d$,

If $L=\{v:n_j\cdot v\le b_j,1\le j\le q\}$ is a polytope, it suffices to check $h_K(n_j)\le b_j$.

Proof. Inclusion gives the support inequalities. If $v\in K\setminus L$, let $z$ be the Euclidean nearest point in $L$. The minimizing property gives $(v-z)\cdot (w-z)\le 0$ for every $w\in L$. Thus the direction $n=v-z$ strictly separates $v$ from $L$, contradicting the support inequalities. The half-space claim is immediate.