Affinely associated bodies

Some of most fundamental objects in mathematics are functions and bodies in ordinary Euclidean space. These are used to represent real objects and phenomena in science and engineering. Here we are interested in properties of these objects that are invariant under linear or affine transformations. Although an affine structure contains no notion of distance or angle, there is still a rich and deep theory that already encompasses many of the most important aspects of Euclidean geometry. The goal of the project is to give a classification of those bodies and functions that are linearly or affinely invariant and have simple additivity properties. This should lead to a better understanding of affine geometry and could prove to be useful in fields ranging from pure mathematical areas such as differential geometry and functional analysis to more applied areas such as information theory and stereology.

Some of most fundamental objects in mathematics are functions and bodies in ordinary Euclidean space. These are used to represent real objects and phenomena in science and engineering. Here we are interested in properties of these objects that are invariant under linear or affine transformations. Since in this context the symmetry group is very large, there are fewer invariant objects and the relations between them are more fundamental. The study of these relations in turn leads to a better understanding of analysis in ordinary Euclidean space. The results obtained during the project show that there is a small class of affinely associated bodies. This underlines their importance and the importance of the inequalities related to them. Using a generalization of these classification results, new inequalities in Euclidean geometry could be obtained. It was shown that beside the applications of affinely associated bodies in fields like functional analysis, stereology and geometric tomography, there are new applications to questions in symbolic dynamics.