Discrete Brunn-Minkowski Inequality for subsets of the cube

We show that for all $A,B\subseteq \{0,1,2{\}}^d$ we have $$\mathrm{\mid }A+B\mathrm{\mid }\ge (\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }{)}^{\log (5)\mathrm{/}(2\log (3))}\mathrm{.}$$ We also show that for all finite $A,B\subset {\mathbb{Z}}^d$, and any $V\subseteq \{0,1{\}}^d$ the inequality $$\mathrm{\mid }A+B+V\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}p}\mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}q}\mathrm{\mid }V{\mathrm{\mid }}^{{\log }_2(p^{1\mathrm{/}p}{q}^{1\mathrm{/}q})}$$ holds for all $p\in (1,\mathrm{\infty })$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. All the estimates are dimension free with the best possible exponents. We discuss applications to various related problems.