  $L_p$ Brunn-Minkowski type inequa\-li\-ties for the lattice point enumerator
$\mathrm{G}_n(\cdot)$ are shown, both in a geometrical and in a functional
setting. In particular, we prove that \[\mathrm{G}_n\bigl((1-\lambda)\cdot K
+_p \lambda\cdot L + (-1,1)^n\bigr)^{p/n}\geq
(1-\lambda)\mathrm{G}_n(K)^{p/n}+\lambda\mathrm{G}_n(L)^{p/n}\] for any $K,
L\subset\mathbb{R}^n$ bounded sets with integer points and all
$\lambda\in(0,1)$. We also show that these new discrete analogues (for
$\mathrm{G}_n(\cdot)$) imply the corresponding results concerning the Lebesgue
measure.
