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 Hörmander’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, Kähler geometry and Algebraic geometry. Bibliographical sketch
Berndtsson has been a member of the Royal Swedish Academy of Sciences since 2003. In 1995 he was awarded the Göran Gustafsson prize.
Berndtsson's first results concern zero sets of holomorphic functions, and in 1981 he showed that any divisor with finite area in the unit ball in the two-dimensional complex space is defined by a bounded holomorphic function (which is not true in higher dimensions). In the 1980s he also developed (together with Mats Andersson) a formalism to generate weighted integral representation formulas for holomorphic functions and solutions to the so-called dbar-equation, which is a the higher-dimensional generalization of the Cauchy-Riemann equations in the plane. This formalism led to new results concerning division and interpolation of holomorphic functions. In the 1990s Berndtsson started to work with L² methods that had been introduced byLars Hörmander, Joseph J. Kohn and others in the 1960s and he modified these methods to obtain uniform estimates for the dbar-equation. At this time he also achieved results about interpolation and sampling in Hilbert spaces of holomorphic functions using L²-estimates.
More recently Berndtsson has worked on global problems on complex manifolds. In a series of papers starting in 2005 he has obtained positivity results for the curvature of holomorphic vector bundles naturally associated to holomorphic fibrations. These vector bundles arise as the zeroth direct images of the adjoint of an ample line bundle over the fibration. The case of a trivial line bundle was considered in earlier work by Phillip Griffiths in connection to variations of Hodge structures and by Fujita, Kawamata and Eckart Viehweg in algebraic geometry. Berndtsson has also explored applications of these positivity results in Kähler geometry (e.g., to geodesics in the space of Kähler metrics) and algebraic geometry (e.g., a new proof of the Kawamata subadjunction formula in a collaboration with Mihai Paun).