The conjectured log-Brunn--Minkowski inequality says that the volume of centrally symmetric convex bodies $K,L\subset\mathbb{R}^n$ satisfies ${vol}\bigl((1-\lambda)\cdot K+_0\lambda\cdot L\bigr) \geq{vol}(K)^{1-\lambda}{vol}(L)^\lambda$, $\lambda\in(0,1)$, and is known to be true in the plane and for particular classes of symmetric convex bodies in $\mathbb{R}^n$. In this paper, we get some discrete log-Brunn--Minkowski type inequalities for the lattice point enumerator. Among others, we show that if $K,L\subset\mathbb{R}^n$ are unconditional convex bodies and $\lambda\in(0,1)$, then ${G}_n((1-\lambda)\cdot(K+C_n)+_0\lambda\cdot(L+C_n)+(-\frac{1}{2},\frac{1}{2})^n) \geq{G}_n(K)^{1-\lambda}{G}_n(L)^\lambda, $ where $C_n=[-1/2,1/2]^n$. Neither $C_n$ nor $(-1/2,1/2)^n$ can be removed. Furthermore, it implies the (volume) log-Brunn--Minkowski inequality for unconditional convex bodies. The corresponding results in the $L_p$ setting for $0

The Brunn--Minkowski inequality states that the volume of compact sets $K,L\subset\mathbb{R}^n$ satisfies $\mathrm{vol}(K+L)^{1/n}\geq\mathrm{vol}(K)^{1/n}+\mathrm{vol}(L)^{1/n}$. In this paper we obtain two discrete analogs of it for the cardinality of finite subsets of the integer lattice ${\mathbb{Z}}^n$. On one hand we prove that if $A,B\subset{\mathbb{Z}}^n$ are finite, then $\bigl|\bar{A}+B\bigr|^{1/n}\geq |A|^{1/n}+|B|^{1/n}$, where $\bar{A}$ is an extension of $A$ which is constructed by adding some new integer points in a particular way; on the other hand, removing points of $A$, say, an equivalent inequality of the form $|A+B|^{1/n}\geq\bigl|{\mathrm{r}}(A)\bigr|^{1/n}+|B|^{1/n}$ can be obtained, where $r(A)$ is the reduced set of $A$. Both inequalities are sharp, and it can be seen that the number of additional points in $\bar{A}$ cannot be too large and depends only on $A$. Finally we also prove that the classical Brunn--Minkowski inequality for compact sets can be obtained as a consequence of these new discrete versions.