  Providing system-size independent lower bounds on the spectral gap of local
Hamiltonian is in general a hard problem. For the case of finite-range,
frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show
that a property of the ground state space is sufficient to obtain such a bound.
We furthermore show that such a condition is necessary and equivalent to a
constant spectral gap. Thanks to this equivalence, we can prove that for
gapless models in any dimension, the spectral gap on regions of diameter $n$ is
at most $o\left(\frac{\log(n)^{2+\epsilon}}{n}\right)$ for any positive
$\epsilon$.
