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| Brunn–Minkowski inequality | |
|---|---|
| Name | Brunn–Minkowski inequality |
| Field | Mathematics |
| Subfield | Convex geometry |
| Introduced | 1887–1910 |
| Contributors | Hermann Brunn; Hermann Minkowski |
Brunn–Minkowski inequality The Brunn–Minkowski inequality is a fundamental result in convex geometry linking the volumes of sets to their Minkowski sum. It plays a central role in the work of many mathematicians and institutions across Europe and appears in the research programs of David Hilbert, Emmy Noether, Felix Klein, Sofia Kovalevskaya, and later researchers at University of Göttingen, University of Leipzig, ETH Zurich, École Normale Supérieure, and Princeton University. The inequality underpins developments in analysis influenced by figures such as Henri Lebesgue, Stefan Banach, John von Neumann, Norbert Wiener, and Paul Erdős.
The classical statement concerns nonempty measurable subsets of Euclidean space and relates Lebesgue measure to Minkowski addition; it was formulated through the work of Hermann Brunn and Hermann Minkowski and subsequently used by Constantin Carathéodory, Ludwig Bieberbach, Erhard Schmidt, Otto Toeplitz, and Georg Cantor in geometric measure theory. Informally, for measurable sets A and B in R^n and t in [0,1], the n-th root of volume is concave under convex combinations; this formulation influenced studies by Andrey Kolmogorov, Pavel Alexandrov, Kazimierz Kuratowski, Issai Schur, and Richard Courant. The inequality connects to major projects in analysis led by Jean Leray, André Weil, Kurt Gödel, Jacques Hadamard, and Eugenio Elia Levi.
The origins trace to late 19th and early 20th-century work of Hermann Brunn and Hermann Minkowski, developed alongside research by Georg Pick, Ludwig Minkowski (relative), Max Dehn, and David Hilbert at conferences paralleling meetings attended by Felix Klein, Bernhard Riemann, Ernst Zermelo, and Gustav Kirchhoff. Subsequent formalizations and proofs were produced by Edmund Landau, Ernst Witt, Stefan Banach, and Hermann Weyl in contexts overlapping programs at University of Cambridge, University of Bonn, University of Paris, and University of Vienna. The inequality later influenced the work of Aleksandr Lyapunov, John von Neumann, Emil Artin, Herman A. Schwartz, and André Weil in both pure and applied directions.
Multiple proofs exist including geometric, analytic, and mass-transport approaches developed by researchers at Cambridge University, Harvard University, Columbia University, and University of Chicago. Classical geometric proofs were advanced by Henri Minkowski (son?) and refined by László Fejes Tóth, Edwin Schrödinger, Marcel Riesz, and Paul Lévy, while analytic proofs invoke inequalities by Henri Lebesgue, Frigyes Riesz, Gábor Szegő, Salomon Bochner, and Elias Stein. Optimal-transport proofs use ideas from Leonid Kantorovich, Ludwig Prange, Cédric Villani, Yann Brenier, and Robert McCann, and functional versions relate to work by Elias Stein, E. T. Bell, John Nash, and Louis de Branges. Tools from measure theory by André Weil, Frigyes Riesz, Maurice Fréchet, and Norbert Wiener appear in many presentations.
Characterizations of equality involve convex bodies and translations studied by Steiner, A. D. Alexandrov, Rolf Nevanlinna, René Thom, and Hermann Weyl; specific rigidity and stability results were established by Donald C. Spencer, Isadore M. Singer, Mikhael Gromov, Mikhail B. Menshov, and Igor Shafarevich. Stability estimates connecting near-equality to near-congruence were developed in work influenced by Paul Erdős, Gábor Szegő, Hugo Steinhaus, and researchers at Moscow State University and University of Warsaw.
The inequality is used across convexity, functional analysis, and information theory in research by Claude Shannon, Norbert Wiener, Andrei Kolmogorov, Paul Lévy, and Alfréd Rényi. It underlies isoperimetric results pursued by Joseph-Louis Lagrange, Siméon Denis Poisson, Carl Friedrich Gauss, Sofia Kovalevskaya, and Henri Poincaré and informs concentration of measure studied by Vitaly Milman, Miklós Révész, Erdős, and Paul Erdős collaborators. Further applications appear in geometric inequalities by A. D. Alexandrov, Aleksandr Danilovich Aleksandrov, Weyl, Eugene Wigner, and in probabilistic limit theorems associated with Andrey Kolmogorov, William Feller, and Ronald Fisher.
Extensions include the Prékopa–Leindler inequality developed by Andras Prekopa, Henryk Leindler, and built upon by Branko Grünbaum, Bálint Szegedy, Cédric Villani, Ludwig Boltzmann-inspired communities, and later generalizations like the Borell–Brascamp–Lieb inequality advanced by Ken Ribet, Boris V. Gnedenko, Christer Borell, Herm Jan Brascamp, and E. H. Lieb. Relations with the isoperimetric inequality, Urysohn inequalities, and Santaló inequality connect to work by Henri Poincaré, J. J. Sylvester, Joseph Liouville, J. A. Schouten, and Marcel Berger. Modern research draws from groups at IHES, MSRI, Clay Mathematics Institute, Institut Henri Poincaré, and American Mathematical Society conferences.
Category:Convex geometry