Harmonic Blaschke-Minkowski Homomorphism

Centroid bodies are a continuous and GL(n)-contravariant valuation and play critical roles in the solution to the Busemann-Petty problem. In this paper, we introduce the notion of harmonic Blaschke-Minkowski homomorphism and show that such a map is represented by a spherical convolution operator. Furthermore, we consider the Shephard-type problem of whether Phi K subset of Phi L implies V (K) <= V (L), where Phi is a harmonic Blaschke-Minkowski homomorphism. Some important results for centroid bodies are extended to a large class of valuations. Finally, we give two interesting results for even and odd harmonic Blaschke-Minkowski homomorphisms, separately.