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 $${\mathrm{G}}_n((1-\lambda )\cdot K{+}_p\lambda \cdot L+(-1,1{)}^n{)}^{p\mathrm{/}n}\ge (1-\lambda ){\mathrm{G}}_n(K{)}^{p\mathrm{/}n}+\lambda {\mathrm{G}}_n(L{)}^{p\mathrm{/}n}$$ for any $K, L\subset$\mathbb{R}$^n$boundedsetswithintegerpointsandall$\lambda\in(0,1)$\mathrm{.}Wealsoshowthatthesenewdiscreteanalogues(for$$\mathrm{G}$_n(\cdot)$) imply the corresponding results concerning the Lebesgue measure.