Harmonic Blaschke–Minkowski Homomorphism

Abstract

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 ΦK⊆ΦL implies V(K)≤V(L), where Φ 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.