A discrete version of the Brunn-Minkowski inequality and its stability

In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for related inequalities (curvature-dimension inequality, metric contraction property) but not for the Brunn-Minkowski one, as far as we know. In the second part of the paper, we show that every metric measure space satisfying the classical Brunn-Minkowski inequality can be approximated by discrete metric spaces with some approximated Brunn-Minkowski inequalities.