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The Brunn–Minkowski inequality gives a lower bound of the Lebesgue measure of a sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory of convex bodies. This topic has many interactions with isoperimetry or functional analysis. Our aim here is to report some recent aspects of these interactions involving optimal mass transport or the Heat equation. Among other things, we will present Brunn–Minkowski inequalities for flat sets, or in Gauss space, as well as local versions of the theorem which apply to the study of entropy production in the central limit theorem.