Valuations and Busemann–Petty type problems

Abstract

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 Φ K ⊆ Φ L 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.