Discrete Brunn-Minkowski Inequality for subsets of the cube

We show that for all subsets A₁, A₂,, Aₙ⊆\0,1\ᵈ, we have |A₁+A₂++Aₙ| ≥ ∏ᵢ₌₁ⁿ|Aᵢ|^(log₂(n+1))/n. In addition, we show that for all A, B ⊆ \0,1,2\ᵈ we have |A+B|≥ (|A||B|)log(5)/(2log(3)). We also show that for all finite A,B ⊂ Zᵈ, and any V ⊆\0,1\ᵈ the inequality |A+B+V|≥ |A|1/p|B|1/q|V|^log₂(p1/pq1/q) holds for all p ∈ (1, ∞), where q=p/p-1 is the conjugate exponent of p. All three estimates are dimension free with the best possible exponents. We also discuss applications to various related problems.