Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester

Let ${\vec {v_0},..., \vec {v_k}}$ be vectors in $\mathbf{Z}^k$which generate $\mathbf{Z}^k$. We show that a body $V \subset \mathbf{Z}^k$ with the vectors ${\vec {v_0},..., \vec {v_k}}$as edge vectors is an almost minimal set with the property that every function $f: V \rightarrow \mathbf{R}$ with periods ${\vec {v_0},..., \vec {v_k}}$ is constant. For $k=1$ the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma.

Suppose $\vec{0}$ is not a non-trivial linear combination of ${\vec {v_0},..., \vec {v_k}}$ with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of ${\vec {v_0},..., \vec {v_k}}$ over $\mathbf{Z}_{\geq 0}$, but infinitely many points on each of its hyperfaces are not. For $k=1$ the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.