  We investigate the asymptotic behaviour of nonlinear Schr\"odinger ground
states on $d$-dimensional periodic metric grids in the limit for the length of
the edges going to zero. We prove that suitable piecewise-affine extensions of
such states converge strongly in $H^1(\mathbb{R}^d)$ to the corresponding
ground states on $\mathbb{R}^d$. As an application of such convergence results,
qualitative properties of ground states and multiplicity results for fixed mass
critical points of the energy on grids are derived. Moreover, we compare
optimal constants in $d$-dimensional Gagliardo-Nirenberg inequalities on
$\mathbb{R}^d$ and on grids. For $L^2$-critical and supercritical powers, we
show that the value of such constants on grids is strictly related to that on
$\mathbb{R}^d$ but, contrary to $\mathbb{R}^d$, constants on grids are not
attained. The proofs of these results combine purely variational arguments with
new Gagliardo-Nirenberg inequalities on grids.
