Research Training Network “ Phenomena in High Dimension ” EDUCATIONAL WORKSHOP ON GEOMETRIC INEQUALITIES In honour of the 65 th birthday

S OF MAIN LECTURES Richard. J. GARDNER (Western Washington Univ.,U.S.A ) The dual Brunn-Minkowski theory and some of its inequalities As the title suggests, the focus of the talk is the dual Brunn-Minkowski theory, initiated by Erwin Lutwak in 1975, and developed since by him and others. First, an elementary introduction is provided to the basics, dual mixed volumes of star bodies, the dual Aleksandrov-Fenchel inequality and some of its consequences. A recent partial extension to bounded Borel sets with a motivating application in local stereology is discussed. After this, intersection bodies are defined and the Busemann intersection inequality is stated in a general form. Finally, affine and dual affine quermassintegrals are introduced and some known and conjectured inequalities involving them are examined. Paul GOODEY (Univ. of Oklahoma, U.S.A ) Classes of centrally symmetric convex bodies Zonoids are highly symmetric convex bodies which arise in many geometric settings. They can be characterized by integral representations of their support functions or in terms of projection bodies. Both these characterizations make use of the lengths of the projections of the body onto lines. We will consider various classes of centrally symmetric bodies that arise from generalizing these characterizations of zonoids to the setting of higher dimensional projections. This will give rise to a number of different generalizations of zonoids. In this talk, we will investigate the connections between these different generalizations. Peter M. GRUBER (Technische Univ. Wien, Austria) John type theorems Using an idea of Voronoi we prove several results characterizing ellipsoids which contain a given convex body and are minimal in some sense. Daniel HUG (Albert Ludwigs Univ. Freiburg, Germany) Valuations, integral geometry and linear dependencies Joint work with R. Schneider and R. Schuster A classical result of integral geometry is the principal kinematic formula, attributed to Blaschke, Santaló and Chern. Another important result, due to Hadwiger, is the characterization theorem for motion invariant, continuous valuations on the space of convex bodies. Both theorems are connected via Hadwiger’s integral geometric formula for general functionals. More recently, a paper by Peter McMullen from 1997 stimulated the investigation of the interplay between tensor valuations and integral geometry. In this direction, a first culmination point was the classification of isometry covariant, continuous valuations by Semyon Alesker. This theorem provides a far reaching generalization of Hadwiger’s theorem on valuations and of the special case concerning vector valuations, which had been considered previously. In this talk, we will start with a brief outline of these developments and then delve into integral geometry of tensor valuations. Some special Crofton and kinematic formulas as well as applications to stochastic geometry have been found. Here we aim at describing a more complete picture of available results and underlying methods. In a final part, we consider the problem of determining all linear dependencies among the basic isometry covariant functionals, a question originally raised by Peter McMullen. Peter McMULLEN (Univ. College London, England) The algebra of polyhedra At the heart of much of the metrical theory of convex bodies and related topics, as well as some of the combinatorial theory of polytopes, lies the concept of a valuation. For polytopes, it turns out to be very useful to develop the initial part of the theory of valuations at the more general level of polyhedra (polyhedral sets). There is then, in fact, a ring structure, which is probably well known to this audience. The addition on the polyhedron ring Γ is induced by the valuation relation: [P ∪Q] + [P ∩Q] = [P ] + [Q]