arXiv subjects
Publications and source records attributed to Thomas Wannerer.
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.
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 classification of all continuous GL(n) equivariant Minkowski valuations on convex bodies in $\mathbb{R}^n$ is established. Together with recent results of F.E. Schuster and the author, this article therefore completes the description of all continuous GL(n) intertwining Minkowski valuations.
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.
The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its Klain function is characterized. Among several applications, it is shown that McMullen's decomposition is not possible in the class of translation-invariant, continuous, positive valuations. This implies that there exists no McMullen decomposition for translation-invariant, continuous Minkowski valuations, which solves a problem first posed by Schneider and Schuster.
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.