Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in $\mathbb{R}^{n+1}$.
In their book "Combinatorial Commutative Algebra" Miller and Sturmfels claim without proof that the Krull dimension of $\mathbb{k}[Q]$ equals the degree of its Hilbert polynomial plus one, where $\mathbb{k}$ is any field and $\mathbb{k}[Q]$ is graded with respect to the $(n+1)$-th coordinate.
Why does this statement hold? Do we really need all these prerequisites here or does it follow from a more general theorem? Note that $\mathbb{k}[Q]$ is a positively graded normal noetherian domain with $0$-homogeneous component $\mathbb{k}$. In particular $\mathbb{k}[Q]$ is a finitely generated $\mathbb{k}$-algebra.