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| Brunn–Minkowski theory | |
|---|---|
| Name | Brunn–Minkowski theory |
| Field | Convex geometry |
| Introduced | 20th century |
| Key figures | Hermann Brunn; Hermann Minkowski; Hans Rademacher; Peter Minkowski |
Brunn–Minkowski theory
Brunn–Minkowski theory is a central area in convex geometry linking geometric measure, additive structure, and functional inequalities. Originating from work by Hermann Brunn and Hermann Minkowski in the early 20th century, the theory connects volume, Minkowski addition, and isoperimetric-type results that influenced research at institutions such as the University of Göttingen, ETH Zurich, and the Institute for Advanced Study. Its development interacted with advances by mathematicians associated with Hilbert's circle, the Bourbaki group, and later researchers at Princeton University, ETH Zurich, and Université Paris-Saclay.
The roots trace to contributions by Hermann Brunn and Hermann Minkowski, who built on geometric studies contemporaneous with figures like David Hilbert, Felix Klein, and Emmy Noether, and influenced later work by John von Neumann, Stefan Banach, and André Weil. Progress in the 20th century involved researchers from University of Göttingen, University of Bonn, University of Vienna, and laboratories such as the Institut des Hautes Études Scientifiques where techniques merged with functional analysis led by scholars like Frigyes Riesz, Marcel Riesz, and Egon Schulte. The mid-century synthesis interfaced with contributions from Paul Lévy, Norbert Wiener, and László Fejes Tóth, and spurred developments in additive combinatorics associated with names like Kurt Mahler, Alfred Rényi, and later Terence Tao and Ben Green.
Key objects include convex bodies studied in the tradition of Johann Kepler's packing problems and formalized by Minkowski, with definitions relying on support functions, mixed volumes, and surface area measures linked to work by Hermann Weyl and Ludwig Bieberbach. Minkowski addition, central to the theory, interacts with notions conceived by Henri Lebesgue and refined through measure theory of Émile Borel and Henri Poincaré. Mixed volume theory was advanced by contributors such as Aleksandr Alexandrov, Hermann Minkowski himself, and later by C. A. Rogers and C. M. Petty, connecting to curvature measures studied by Marcel Berger and Shing-Tung Yau.
The Brunn–Minkowski inequality expresses a concavity property of volume under Minkowski addition, a theme present in the legacies of Hermann Minkowski, Hermann Brunn, and later expositors like Rolf Schneider and Vitali Milman. Functional forms, including Prekopa–Leindler and Borell inequalities, were developed by Prékopa and Leindler and related to work by Christer Borell and Richard Borell's contemporaries. Extensions and alternative formulations were influenced by probabilists such as Paul Lévy and analysts like Elias Stein and László Lovász, with convexity methods adopted in research at Massachusetts Institute of Technology, Courant Institute, and IHÉS.
Applications permeate study of isoperimetric problems historically pursued by Joseph-Louis Lagrange and Sophie Germain, and modern treatments by Isaac Newton's successors in geometric optics and packing. Results impact the solution of optimization problems linked to John Nash and variational methods from Richard Courant's school, and inform algorithms developed at institutions such as Bell Labs and Bellcore for computational geometry. Connections extend to analytic inequalities used by Elias Stein, probabilistic limit theorems studied by Andrey Kolmogorov, and functional analysis trends from Stefan Banach and John von Neumann.
Research on stability and equality cases built on classical rigidity themes present in the work of Aleksandr Alexandrov and Ludwig Bieberbach, with modern quantitative stability results contributed by researchers tied to Tel Aviv University and University of Washington. Extensions include Aleksandrov–Fenchel type inequalities influenced by Werner Fenchel and A.D. Aleksandrov, and developments in geometric tomography pursued by scholars at University of Cambridge and University of Oxford. Recent quantitative frameworks draw on techniques from additive combinatorics popularized by Tim Gowers and Ben Green.
The theory interrelates with classical inequalities such as the isoperimetric inequality studied by Joseph Plateau and Lord Kelvin, Sobolev inequalities refined by Sergei Sobolev and Laurent Schwartz, and the Hölder and Minkowski inequalities rooted in the work of Otto Hölder and Hermann Minkowski. Generalizations span functional inequalities like Prekopa–Leindler (connected to András Prékopa and János Leindler), and geometric inequalities informed by the Alexandrov–Fenchel theory and contributions from Henri Lebesgue-era measure theory, with ongoing advances at centres such as Princeton University, ETH Zurich, IHÉS, and Institut Mittag-Leffler.
Category:Convex geometry