The talk is devoted to a classical subject in the fields of convex geometry and functional inequalities: the Brunn-Minkowski inequality. In simple terms, it relates the volume of the (Minkowski) sum of two compact sets with the sum of the volumes of the sets. In the talk two proofs of the Brunn-Minkowski inequality will be shown: the first one is a geometric proof based on the study of boxes and the second one is a more analytical proof (based on the so-called Prekopa-Leinder inequality). Then, the fact that the Brunn-Minkowski inequality quickly yields to the isoperimetric one will be presented. Also the case of more general variational functional (e.g. the first Dirichlet eigenvalue of the Laplace operator, the capacity, and the torsional rigidity) will be discussed in the talk. Finally, if time permits, the recent and so-called Lp-Brunn-Minkowski inequality conjectured by Böröczky, Lutwak, Yang, and Zhang will be presented.
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