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We give an alternative proof for discrete Brunn-Minkowski type inequalities,recently obtained by Halikias, Klartag and the author. This proof also impliessomewhat stronger weighted versions of these inequalities. Our approachgeneralizes ideas of Gozlan, Roberto, Samson and Tetali from the theory ofmeasure transportation and provides new displacement convexity of entropy typeinequalities for the lattice point enumerator.

A Remark on discrete Brunn-Minkowski type inequalities via transportation of measure