Randomized isoperimetric inequalities for Lp-centroid bodies

Abstract

This thesis is concerned with the Lp-Busemann-Petty centroid inequality, an affine isoperimetric inequality, which compares the volume of a convex body in n-dimensional Euclidean space with that of its Lp-centroid body as an extension of the classical Busemann-Petty centroid inequality. Isoperimetric-type inequalities not only occupy a central role in the field of geometric convexity but also have numerous applications to fields such as ordinary and partial differential equations, functional analysis, the geometry of numbers, discrete geometry and polytopal approximations, stereology and stochastic geometry, and Minkowskian geometry. On the one hand, we present a direct proof of the Lp-Busemann-Petty centroid inequality by Campiand Gronchi which does not use the Lp-analog of the Petty projection inequality but instead uses shadow systems. On the other hand, we present a randomized version of the same inequality due to Paouris and Pivovarov using an extension of Groemer’s theorem to the class of all probability measures that are absolutely continuous with respect to Lebesgue measure and rearrangement inequalities. Additionally, we present a randomized version of the polar Lp-Busemann-Petty centroid inequality due to Cordero-Erausquin, Fradelizi, Paouris and Pivovarov which combines methods and ideas from both topics above.