As a generalization of the notion of measure, valuations on convex sets have always played a central role in geometry. A particularly exciting new development in the theory of valuations explores the strong connections between convex body valued valuations and the theory of affine isoperimetric and analytic inequalities. To be more specific, many powerful affine isoperimetric inequalities involve linearly intertwining Minkowski valuations. Although a large part of the theory of convex body valued valuations deals with operators compatible with linear transformations, considerable effort has been invested in recent years to also classify all continuous rigid motion compatible Minkowski valuations. These results in turn have applications to Brunn-Minkowski type inequalities which were shown to hold for various large classes of valuations equivariant under orthogonal transformations. A goal of this project is to further clarify which of the classical (and more recent) affine geometric inequalities can be generalized to functionals derived from Minkowski valuations which are compatible with orthogonal transformations only. Analytic descriptions of these Minkowski valuations will play a key role in these efforts. The theory of valuations is deeply intertwined with the Brunn-Minkowski theory which arises from combining the notion of Minkowski addition of convex bodies with that of ordinary volume. Two decades ago, a combination of an old notion of Minkowski-Firey Lp addition with volume led to an embryonic Lp Brunn-Minkowski theory. Over the last 20 years this rapidly growing theory has built strong ties with the theory of valuations. For example an Lp analog of the classical projection operator was introduced and an important Lp extension of one of the fundamental affine isoperimetric inequalities, the Petty projection inequality, was established. This extension is the core of a sharp affine Lp Sobolev inequality which strengthens the classical Lp Sobolev inequality. However, the advances in valuation theory revealed that the Lp projection operator is only one representative of an entire class of Lp extensions of the classical projection operator. This fundamental result has very recently led to a further generalization of Petty's projection inequality and a new asymmetric affine Lp Sobolev inequality. The well known equivalence of the isoperimetric inequality and the sharp Sobolev inequality is an important example of the interplay between analytic and geometric inequalities. This remarkable link has been amplified by the recent work on affine analytic inequalities. It is one of the aims of this project to further exploit these strong relations and to establish new affine log-Sobolev and Gagliardo-Nirenberg inequalities.