The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the algebra associated to this co-product is described explicitly in terms of generators and relations. As a consequence, a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained.

A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.

Abstract. The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.

A convolution representation of continuous translation invariant and SO(n) equivariant Minkowski valuations is established. This is based on a new classification of translation invariant generalized spherical valuations. As applications, Crofton and kinematic formulas for Minkowski valuations are obtained., the subspace of SO(n − 1) invariant valuations in Val i . In turn, valuations in Val SO(n−1) i are spherical.

Abstract.A new integral representation of smooth translation invariant and rotation equivariant even Minkowski valuations is established. Explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived. Moreover, the action of Alesker's Hard Lefschetz operators on these Minkowski valuations is explored in detail.