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Geometric and functional Brunn-Minkowski type inequalities for the latticepoint enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we showthat $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq(1-\lambda)\mathrm{G}_n(K)^{1/n}+\lambda\mathrm{G}_n(L)^{1/n}$$ for anynon-empty bounded sets $K, L\subset\mathbb{R}^n$ and all $\lambda\in(0,1)$.  We also show that these new discrete versions imply the classical results,and discuss some links with other related inequalities.

Brunn-Minkowski type inequalities for the lattice point enumerator