Volume inequalities for Minkowski valuations

Abstract

In this thesis, new Orlicz-Brunn-Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. This is a joint work with Lukas Parapatits, Franz Schuster and Manuel Weberndorfer. The second focus of this thesis lies on the generalization of Lutwak's volume inequalities for polar projection bodies of all orders to polarizations of Minkowski valuations generated by o-symmetric zonoids of revolution. This is based on generalizations of the notions of centroid bodies and mixed projection bodies to such Minkowski valuations. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of Leng and Lu's volume inequalities for intersection bodies of all orders is proved. These results are related to Grinberg's inequalities for dual affine quermassintegrals.