In certain special cases of later interest we can describe $Q^{\lambda}_{(I\mid J)}(0)$ as follows:

1. (1)

   If $J=\emptyset$, then $Q^{\lambda}_{I}(0)$ is non-empty if and only if $I$ is the set of vertices of some simplex $\overline{\sigma}\in S\ast\{0\}$, in which case $Q^{\lambda}_{I}(0)=Q^{\lambda}_{\overline{\sigma}}$.

2. (2)

   $Q^{\lambda}_{(\{0\}\mid v)}(0)$ contains the relative interior of the facet $G_{v}\subset\Delta^{\vee}_{\lambda}$.

3. (3)

   $Q^{\lambda}_{(\{0\}\mid w^{\perp})}(0)$ is the Minkowski sum of the polytope $\Delta^{\vee}_{\lambda}$ and the normal cone to $\operatorname{carrier}_{\Delta\backslash w^{\perp}}\{0\}$:

   $$Q^{\lambda}_{(\{0\}\mid w^{\perp})}(0)=\Delta^{\vee}_{\lambda}+\operatorname{NC}_{\Delta\backslash w^{\perp}}(\{0\}).$$

   In particular, it contains $\Delta^{\vee}_{\lambda}+\operatorname{cone}(w)$.