Geometric and functional inequalities (Q6612728)

This article is part of the book [Convex geometry. Cetraro, Italy, August 30 -- September 3, 2021. Lecture notes given at the summer school. Cham: Springer; Florence: Fondazione CIME (2023; Zbl 1530.52001)], edited by \textit{A. Colesanti} (ed.) and \textit{M. Ludwig} (ed.).\N\NIn this chapter the authors provide a general nice overview of the field of geometric and functional inequalities, describing some of the main relations for the basic geometric functionals on convex bodies.\N\NThey start showing two relevant inequalities in convex geometry: the Brunn-Minkowski and the Aleksandrov-Fenchel inequalities, providing several proofs of them as well as some applications. Next, they move to the so-called \(L_p\)-Brunn-Minkowski theory, and present several meaningful results, as the \(L_p\) version of the Brunn-Minkowski inequality, \(p\geq 1\), or the beautiful conjectured log-Brunn-Minkowski inequality. In the next section, Brunn-Minkowski type inequalities for the electrostatic capacity, the torsion and the first Dirichlet eigenvalue of the Laplace operator are discussed. Finally, the major Brascamp-Lieb and Barthe inequalities are shown, including some of their main geometric applications as the volume ratio inequality or the reverse isoperimetric inequality.\N\NFor the entire collection see [Zbl 1530.52001].