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Explore fundamental geometric inequalities in this graduate-level mathematics lecture from the Instituto de Matemática Pura e Aplicada's Summer School 2026. Delve into mean inequalities, Hölder, Minkowski, and Jensen inequalities, along with Stolarsky means and Palés inequality, culminating in Khintchine's theorem. Master rearrangement inequalities and their applications including Riesz, Chebyshev, and Hardy-Littlewood-Pólya inequalities. Examine the equivalence of majorizations, Schur convexity, symmetric functions, and Muirhead's theorem. Study Hilbert and Hardy inequalities, Wirtinger's inequality, and Steiner symmetrization. Investigate Hardy-Littlewood-Sobolev inequalities with applications, and conclude with the Brunn-Minkowski inequality and its consequences including isoperimetric and Prékopa-Leindler inequalities. Learn from professors João Pedro Ramos and Felipe Gonçalves as they present rigorous mathematical foundations essential for advanced study in geometric analysis and optimization theory.
