Let $A \subseteq \mathbb{Z}^m$. A point $\mathbf{a} \in A$ is an [**extreme point**]{} if $\mathbf{a} =  t\mathbf{b}+(1-t)\mathbf{c}$ for some $\mathbf{b}, \mathbf{c} \in A$ and $t\in (0,1)$ implies $\mathbf{b}=\mathbf{c}=\mathbf{a}$.

By Minkowski’s Theorem a compact convex set is a convex hull of the set of its extreme points. Thus, ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ is the convex hull of its extreme points.

\[proposition\_vertices\_of\_sgp\] Let $h$ be a positive integer, and let $\lambda$ be a partition with at most $m$ parts. Suppose $\deg G_{h,\lambda}(\x) = |\lambda|+N$, and let $\lambda^{(0)},\ldots, \lambda^{(N)}$ be the sequence of dominating partitions for $G_{h,\lambda}(\x)$. Suppose $\lambda^{(N)} =(\lambda_1+a_1, \ldots, \lambda_m+a_m)$ for some nonnegative integers $a_1,\ldots, a_m$, and let $b_k = a_1+\cdots+ a_k$ for $k=1,\ldots ,m$. Then $${\mathrm{Newt}}(G_{h,\lambda}(\x)) = \mathrm{conv} \bigcup_{k=1}^m {\mathrm{Newt}}(s_{\lambda^{(b_k)}}(\x))$$ is the convex hull of the $S_m$-orbits of the partitions $\lambda^{(b_1)}, \ldots, \lambda^{(b_m)}$. Moreover, $\lambda^{(b_1)} = \lambda^{(0)} =\lambda$ and $\lambda^{(b_m)} = \lambda^{(N)}$.

We shall show that for $k=0,\ldots, N$, $\lambda^{(k)}$ is an extreme point of ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ only if $k\in \{b_1,\ldots, b_m\}$.

If $\p=(p_1,\ldots,p_m)$ is an extreme point of ${\mathrm{Newt}}(G_{h,\lambda}(\x))$, then any permutation of $\p$ is also an extreme point of ${\mathrm{Newt}}(G_{h,\lambda}(\x))$, since ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ is the convex hull of the $S_m$-orbits of $
\lambda^{(0)},\ldots,\lambda^{(N)}$ by Proposition \[proposition\_layers\_of\_polytope\].

As $\lambda^{(N)} = (\lambda_1+a_1, \ldots, \lambda_m+a_m),$ then 