Equivariant endomorphisms of the space of convex bodies

Abstract

We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if d = 2 d = 2 . For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.