Valuations and Busemann–Petty type problems

Projection and intersection bodies define continuous and GL(n) contravariant valuations. They played a critical role in the solution of the Shephard problem for projections of convex bodies and its dual version for sections, the Busemann–Petty problem. We consider the question whether ΦK⊆ΦL implies V(K)⩽V(L), where Φ is a homogeneous, continuous operator on convex or star bodies which is an SO(n) equivariant valuation. Important previous results for projection and intersection bodies are extended to a large class of valuations.