On discrete $L_p$ Brunn-Minkowski type inequalities

$L_p$ 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.