This is an overview of some of the topics mentioned above: Abstract：The classical Brunn-Minkowski inequality is an inequality for volumes of convex sets. It has become the cornerstone of a whole branch of mathematics, called ’Convex Geometry’, and also found many important applications in other fields like harmonic analysis, geometry of Banach spaces and probability. To quote from R Gardner’s survey article in the BAMS (2002):“In a sea of mathematics, the Brunn-Minkowski inequality appears like an octopus, tentacles reaching far and wide...”.

    There are many quite different proofs of the theorem, including some very simple and elementary. A very interesting proof was found by Brascamp and Lieb in 1976. It is based on a weighted L2-estimate for the equation du = f . This inequality of Brascamp and Lieb can be seen as the real variable analog of H?rmander’s L2-estimate for the -equation, and it is therefore natural to ask what inequalities would result from an argument a la Brascamp and Lieb in the complex setting. The basic objects of study are then not volumes of sets, but L2-norms of holomorphic functions, forms, or sections of line bundles - think of the volume of a set as the squared L2-norm of the function 1 defined on that set. The resulting theory gives new inequalities for volumes of sets, just like the classical Brunn-Minkowski inequality, but also has interesting applications in Complex analysis, Kahler geometry and Algebraic geometry.