Here, $\Delta^{\vee}$ is a Minkowski summand of $\Delta^{\vee}_{\lambda}$. Let $\psi$ be a piecewise linear concave function with domains of linearity given by $\operatorname{bsd}(\Delta^{\vee}_{\lambda};\Delta^{\vee})$. Now $\nu$ induces another piecewise linear (non-concave) function $\tilde{\nu}$ on $\Delta^{\vee}_{\lambda}$, which is $\nu$ in $G$-direction on $(F_{0}\prec F_{2}\prec\cdots\prec F_{r},G)$, and constant in $F_{i}$-direction.

The function $\psi$ is strictly concave wherever $\tilde{\nu}$ is non-concave, so that for large $N$, the function $N\psi+\tilde{\nu}$ will be concave. Its domains of linearity are products of simplices that correspond to simplices of $\operatorname{bsd}(\Delta^{\vee}_{\lambda})\cong\operatorname{bsd}(S)$ times simplices of $T$. ∎