Invariant smooth valuations on the Euclidean unit sphere

Abstract

Valuations on the Euclidean unit sphere are additive maps from the set of spherical convex bodies to real numbers. In this thesis we focus on the subspace of smooth valuations. These can be represented by integration of differential forms over so-called normal cycles - sets that generalize the graph of the Gauss map to convex bodies with non-smooth boundary. Using the theory of valuations on arbitrary smooth manifolds developed by S. Alesker et al, we show that the space of smooth spherical valuations that are invariant under the natural action of the special orthogonal group is finite-dimensional. Moreover, a basis of this space is given by the spherical intrinsic volumes. We also obtain a classification of invariant generalized valuations, which form the topological dual space to the space of smooth valuations. Finally, we present a method due to J. Fu of transferring integral geometric formulas from Euclidean space to the sphere, which also yields the above results.