For $I$ and $J$, two disjoint collections of integral points in $\Delta$, and a real number $\epsilon\geq 0$, we define the (possibly empty) polyhedron $Q^{\lambda}_{(I\mid J)}(\epsilon)$ in $\mathbb{R}^{d}$ by the conditions:

	$$\displaystyle\langle m^{\prime\prime},n\rangle+\lambda(m^{\prime\prime})<\langle m,n\rangle+\lambda(m)-\epsilon,$$
	$$\displaystyle\langle m^{\prime},n\rangle+\lambda(m^{\prime})=\langle m,n\rangle+\lambda(m),$$

for all $m,m^{\prime}\in I,m^{\prime\prime}\not\in I\cup J$. We will abbreviate $Q^{\lambda}_{(I\mid J)}(\epsilon)$ by $Q^{\lambda}_{I}(\epsilon)$ when $J$ is empty.