Brunn-Minkowski type inequalities for the lattice point enumerator

Abstract

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator ${\mathrm{G}}_n(\cdot )$ are provided. In particular, we show that $${\mathrm{G}}_n((1-\lambda )K+\lambda L+(-1,1{)}^n{)}^{1/n}\ge (1-\lambda ){\mathrm{G}}_n(K{)}^{1/n}+\lambda {\mathrm{G}}_n(L{)}^{1/n}$$ for any non-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.

Corrected typos. Main results unchanged