We generalize the existing finite-size criteria for spectral gaps of
frustration-free spin systems to D>2 dimensions. We obtain a local gap
threshold of n3, independent of D, for nearest-neighbor
interactions. The n1 scaling persists for arbitrary finite-range
interactions in Z3. The key observation is that there is more
flexibility in Knabe's combinatorial approach if one employs the operator
Cauchy-Schwarz inequality.Comment: 16 page