Stability in the Busemann–Petty and Shephard problems

A comparison problem for volumes of convex bodies asks whether inequalities fK(ξ)⩽fL(ξ) for all ξ∈Sn−1 imply that Voln(K)⩽Voln(L), where K, L are convex bodies in Rn, and fK is a certain geometric characteristic of K. By linear stability in comparison problems we mean that there exists a constant c such that for every ε>0, the inequalities fK(ξ)⩽fL(ξ)+ε for all ξ∈Sn−1 imply that .We prove such results in the settings of the Busemann–Petty and Shephard problems and their generalizations. We consider the section function fK(ξ)=SK(ξ)=Voln−1(K∩ξ⊥) and the projection function fK(ξ)=PK(ξ)=Voln−1(K|ξ⊥), where ξ⊥ is the central hyperplane perpendicular to ξ, and K|ξ⊥ is the orthogonal projection of K to ξ⊥. In these two cases we prove linear stability under additional conditions that K is an intersection body or L is a projection body, respectively. Then we consider other functions fK, which allow to remove the additional conditions on the bodies in higher dimensions.