For (1) we notice that the set of defining inequalities of $Q^{\lambda}_{I}(0)$ is exactly the condition that the maximum in $L_{\lambda}$ is saturated by the linear functions $\langle m,n\rangle+\lambda(m),\ m\in I$.

For (2), note that $Q^{\lambda}_{v\ast\{0\}}$ is defined by the same inequalities as $Q^{\lambda}_{(\{0\}\mid v)}(0)$, plus an extra condition: $\langle v,n\rangle+\lambda(v)=\lambda(0)$.

For (3) we can study the Legendre transform of the restriction of $\lambda$ to the truncated polytope $\Delta\backslash w^{\perp}$ (which is still a concave function). From this point of view, the Minkowski sum in (3) is in complete analogy with (2) of Lemma 3.1.

Note that in the process of truncation we removed all integral points of $\Delta$ with $\langle m,w\rangle=1$. Hence, the remaining ones satisfy $\langle m,w\rangle\leq 0$ which implies that $\{0\}$ is on the boundary of $\Delta\backslash w^{\perp}$, and $w$ is in (the boundary of) the normal cone $\operatorname{NC}_{\Delta\backslash w^{\perp}}(\{0\})$. ∎