Brunn-Minkowski type inequalities for the lattice point enumerator

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

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