Multi-dimensional versions of a theorem of fine and wilf and a formula of Sylvester

Let v 0 ,..., v k be vectors in Z k which generate Z k . We show that a body V ⊂ Z k with the vectors v 0 ,..., v k as edge vectors is an almost minimal set with the property that every function f: V → R with periods v 0 ,..., 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 0 is not a non-trivial linear combination of v 0 ,..., v k with nonnegative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of v 0 ,..., v k over Z ≥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.